I'm aware that this is not an answer but I can't post this error output using a comment.
Below my error output of your 2. error. I got it also when I run in Dartium after I couldn't reproduce the 1. error anymore.
It seems Angular is somehow processing code of the paper-elements.
The error I now get is:
No getter for '$'.
STACKTRACE:
#0 StaticClosureMap.lookupGetter (package:angular/core/parser/parser_static.dart:15:25)
#1 DynamicParserBackend.newAccessScope (package:angular/core/parser/dynamic_parser.dart:108:38)
#2 DynamicParserImpl.parseAccessOrCallScope (package:angular/core/parser/dynamic_parser_impl.dart:257:67)
#3 DynamicParserImpl.parsePrimary (package:angular/core/parser/dynamic_parser_impl.dart:239:36)
#4 DynamicParserImpl.parseAccessOrCallMember (package:angular/core/parser/dynamic_parser_impl.dart:193:30)
#5 DynamicParserImpl.parsePrefix (package:angular/core/parser/dynamic_parser_impl.dart:188:37)
#6 DynamicParserImpl.parseMultiplicative (package:angular/core/parser/dynamic_parser_impl.dart:163:29)
#7 DynamicParserImpl.parseAdditive (package:angular/core/parser/dynamic_parser_impl.dart:149:37)
#8 DynamicParserImpl.parseRelational (package:angular/core/parser/dynamic_parser_impl.dart:131:31)
#9 DynamicParserImpl.parseEquality (package:angular/core/parser/dynamic_parser_impl.dart:117:33)
#10 DynamicParserImpl.parseLogicalAnd (package:angular/core/parser/dynamic_parser_impl.dart:108:31)
#11 DynamicParserImpl.parseLogicalOr (package:angular/core/parser/dynamic_parser_impl.dart:99:33)
#12 DynamicParserImpl.parseConditional (package:angular/core/parser/dynamic_parser_impl.dart:83:32)
#13 DynamicParserImpl.parseExpression (package:angular/core/parser/dynamic_parser_impl.dart:68:34)
#14 DynamicParserImpl.parseFormatter (package:angular/core/parser/dynamic_parser_impl.dart:53:33)
#15 DynamicParserImpl.parsePrimary (package:angular/core/parser/dynamic_parser_impl.dart:220:34)
#16 DynamicParserImpl.parseAccessOrCallMember (package:angular/core/parser/dynamic_parser_impl.dart:193:30)
#17 DynamicParserImpl.parsePrefix (package:angular/core/parser/dynamic_parser_impl.dart:188:37)
#18 DynamicParserImpl.parseMultiplicative (package:angular/core/parser/dynamic_parser_impl.dart:163:29)
#19 DynamicParserImpl.parseAdditive (package:angular/core/parser/dynamic_parser_impl.dart:152:67)
#20 DynamicParserImpl.parseRelational (package:angular/core/parser/dynamic_parser_impl.dart:131:31)
#21 DynamicParserImpl.parseEquality (package:angular/core/parser/dynamic_parser_impl.dart:117:33)
#22 DynamicParserImpl.parseLogicalAnd (package:angular/core/parser/dynamic_parser_impl.dart:108:31)
#23 DynamicParserImpl.parseLogicalOr (package:angular/core/parser/dynamic_parser_impl.dart:99:33)
#24 DynamicParserImpl.parseConditional (package:angular/core/parser/dynamic_parser_impl.dart:83:32)
#25 DynamicParserImpl.parseExpression (package:angular/core/parser/dynamic_parser_impl.dart:68:34)
#26 DynamicParserImpl.parseFormatter (package:angular/core/parser/dynamic_parser_impl.dart:53:33)
#27 DynamicParserImpl.parseChain (package:angular/core/parser/dynamic_parser_impl.dart:35:32)
#28 DynamicParser._parse (package:angular/core/parser/dynamic_parser.dart:36:46)
#29 DynamicParser.call.<anonymous closure> (package:angular/core/parser/dynamic_parser.dart:31:50)
#30 _HashMap.putIfAbsent (dart:collection-patch/collection_patch.dart:124)
#31 DynamicParser.call (package:angular/core/parser/dynamic_parser.dart:31:30)
#32 _AstParser.call (package:angular/core/scope.dart:1063:24)
#33 Scope.watch (package:angular/core/scope.dart:242:35)
#34 TextMustache.TextMustache (package:angular/core_dom/mustache.dart:15:16)
#35 ElementBinder._createDirectiveFactories.<anonymous closure> (package:angular/core_dom/element_binder.dart:244:20)
#36 FactoryProvider.get (package:di/src/provider.dart:49:15)
#37 BaseInjector.getInstanceByKey (package:di/src/base_injector.dart:117:29)
#38 BaseInjector.getByKey (package:di/src/base_injector.dart:162:23)
#39 ElementBinder._link.<anonymous closure> (package:angular/core_dom/element_binder.dart:206:44)
#40 List.forEach (dart:core-patch/growable_array.dart:228)
#41 ElementBinder._link (package:angular/core_dom/element_binder.dart:205:33)
#42 ElementBinder.bind (package:angular/core_dom/element_binder.dart:324:10)
#43 TaggingViewFactory._bindTagged (package:angular/core_dom/tagging_view_factory.dart:43:31)
#44 TaggingViewFactory._link (package:angular/core_dom/tagging_view_factory.dart:76:22)
#45 TaggingViewFactory.call (package:angular/core_dom/tagging_view_factory.dart:22:12)
#46 Application.run.<anonymous closure>.<anonymous closure> (package:angular/application.dart:172:22)
#47 _rootRunUnary (dart:async/zone.dart:842)
#48 _ZoneDelegate.runUnary (dart:async/zone.dart:466)
#49 _onRunUnary.<anonymous closure> (package:angular/core/zone.dart:113:63)
#50 VmTurnZone._onRunBase (package:angular/core/zone.dart:97:16)
#51 _onRunUnary (package:angular/core/zone.dart:113:17)
#52 _CustomZone.runUnary (dart:async/zone.dart:748)
#53 _Future._propagateToListeners.handleValueCallback (dart:async/future_impl.dart:488)
#54 _Future._propagateToListeners (dart:async/future_impl.dart:571)
#55 _Future._completeWithValue (dart:async/future_impl.dart:331)
#56 _Future._asyncComplete.<anonymous closure> (dart:async/future_impl.dart:393)
#57 _rootRun (dart:async/zone.dart:835)
#58 _ZoneDelegate.run (dart:async/zone.dart:459)
#59 _onScheduleMicrotask.<anonymous closure> (package:angular/core/zone.dart:117:43)
#60 VmTurnZone._finishTurn (package:angular/core/zone.dart:143:34)
#61 VmTurnZone._onRunBase (package:angular/core/zone.dart:104:43)
#62 _onRun (package:angular/core/zone.dart:109:17)
#63 _CustomZone.run (dart:async/zone.dart:740)
#64 VmTurnZone.run (package:angular/core/zone.dart:221:40)
#65 Application.run (package:angular/application.dart:164:20)
#66 main.<anonymous closure>.<anonymous closure> (http://localhost:8080/Application.dart:26:14)
#67 _RootZone.runUnary (dart:async/zone.dart:1088)
#68 _Future._propagateToListeners.handleValueCallback (dart:async/future_impl.dart:488)
#69 _Future._propagateToListeners (dart:async/future_impl.dart:571)
#70 _Future._completeWithValue (dart:async/future_impl.dart:331)
#71 _Future._asyncComplete.<anonymous closure> (dart:async/future_impl.dart:393)
#72 _asyncRunCallbackLoop (dart:async/schedule_microtask.dart:23)
#73 _asyncRunCallback (dart:async/schedule_microtask.dart:32)
#74 _handleMutation (dart:html:38347)
The input field at #3 has the value
"\n\n Polymer('core-selector', {\n\n /**\n * Gets or sets the selected element. Default to use the index\n * of the item element.\n *\n * If you want a specific attribute value of the element to be\n * used instead of index, set \"valueattr\" to that attribute name.\n *\n * Example:\n *\n * <core-selector valueattr=\"label\" selected=\"foo\">\n * <div label=\"foo\"></div>\n * <div label=\"bar\"></div>\n * <div label=\"zot\"></div>\n * </core-selector>\n *\n * In multi-selection this should be an array of values.\n *\n * Example:\n *\n * <core-selector id=\"selector\" valueattr=\"label\" multi>\n * <div label=\"foo\"></div>\n * <div label=\"bar\"></div>\n * <div label=\"zot\"></div>\n * </core-selector>\n *\n * this.$.selector.selected = ['foo', 'zot'];\n *\n * @attribute selected\n * @type Object\n * @default null\n */\n selected: null,\n\n /**\n * If true, multiple selections are allowed.\n *\n * @attribute multi\n * @type boolean\n * @default false\n */\n multi: false,\n\n /**\n * Specifies the attribute to be used for \"selected\" attribute.\n *\n * @attribute valueattr\n * @type string\n * @default 'name'\n */\n valueattr: 'name',\n\n /**\n * Specifies the CSS class to be used to add to the selected element.\n * \n * @attribute selectedClass\n * @type string\n * @default 'core-selected'\n */\n selectedClass: 'core-selected',\n\n /**\n * Specifies the property to be used to set on the selected element\n * to indicate its active state.\n *\n * @attribute selectedProperty\n * @type string\n * @default ''\n */\n selectedProperty: '',\n\n /**\n * Specifies the attribute to set on the selected element to indicate\n * its active state.\n *\n * @attribute selectedAttribute\n * @type string\n * @default 'active'\n */\n selectedAttribute: 'active',\n\n /**\n * Returns the currently selected element. In multi-selection this returns\n * an array of selected elements.\n * \n * @attribute selectedItem\n * @type Object\n * @default null\n */\n selectedItem: null,\n\n /**\n * In single selection, this returns the model associated with the\n * selected element.\n * \n * @attribute selectedModel\n * @type Object\n * @default null\n */\n selectedModel: null,\n\n /**\n * In single selection, this returns the selected index.\n *\n * @attribute selectedIndex\n * @type number\n * @default -1\n */\n selectedIndex: -1,\n\n /**\n * The target element that contains items. If this is not set \n * core-selector is the container.\n * \n * @attribute target\n * @type Object\n * @default null\n */\n target: null,\n\n /**\n * This can be used to query nodes from the target node to be used for \n * selection items. Note this only works if the 'target' property is set.\n *\n * Example:\n *\n * <core-selector target=\""+($.myForm|stringify)+"\" itemsSelector=\"input[type=radio]\"></core-selector>\n * <form id=\"myForm\">\n * <label><input type=\"radio\" name=\"color\" value=\"red\"> Red</label> <br>\n * <label><input type=\"radio\" name=\"color\" value=\"green\"> Green</label> <br>\n * <label><input type=\"radio\" name=\"color\" value=\"blue\"> Blue</label> <br>\n * <p>color = "+(color|stringify)+"</p>\n * </form>\n * \n * @attribute itemSelector\n * @type string\n * @default ''\n */\n itemsSelector: '',\n\n /**\n * The event that would be fired from the item element to indicate\n * it is being selected.\n *\n * @attribute activateEvent\n * @type string\n * @default 'tap'\n */\n activateEvent: 'tap',\n\n /**\n * Set this to true to disallow changing the selection via the\n * `activateEvent`.\n *\n * @attribute notap\n * @type boolean\n * @default false\n */\n notap: false,\n\n ready: function() {\n this.activateListener = this.activateHandler.bind(this);\n this.observer = new MutationObserver(this.updateSelected.bind(this));\n if (!this.target) {\n this.target = this;\n }\n },\n\n get items() {\n if (!this.target) {\n return [];\n }\n var nodes = this.target !== this ? (this.itemsSelector ? \n this.target.querySelectorAll(this.itemsSelector) : \n this.target.children) : this.$.items.getDistributedNodes();\n return Array.prototype.filter.call(nodes || [], function(n) {\n return n && n.localName !== 'template';\n });\n },\n\n targetChanged: function(old) {\n if (old) {\n this.removeListener(old);\n this.observer.disconnect();\n this.clearSelection();\n }\n if (this.target) {\n this.addListener(this.target);\n this.observer.observe(this.target, {childList: true});\n this.updateSelected();\n }\n },\n\n addListener: function(node) {\n node.addEventListener(this.activateEvent, this.activateListener);\n },\n\n removeListener: function(node) {\n node.removeEventListener(this.activateEvent, this.activateListener);\n },\n\n get selection() {\n return this.$.selection.getSelection();\n },\n\n selectedChanged: function() {\n this.updateSelected();\n },\n\n updateSelected: function() {\n this.validateSelected();\n if (this.multi) {\n this.clearSelection();\n this.selected && this.selected.forEach(function(s) {\n this.valueToSelection(s);\n }, this);\n } else {\n this.valueToSelection(this.selected);\n }\n },\n\n validateSelected: function() {\n // convert to an array for multi-selection\n if (this.multi && !Array.isArray(this.selected) && \n this.selected !== null && this.selected !== undefined) {\n this.selected = [this.selected];\n }\n },\n\n clearSelection: function() {\n if (this.multi) {\n this.selection.slice().forEach(function(s) {\n this.$.selection.setItemSelected(s, false);\n }, this);\n } else {\n this.$.selection.setItemSelected(this.selection, false);\n }\n this.selectedItem = null;\n this.$.selection.clear();\n },\n\n valueToSelection: function(value) {\n var item = (value === null || value === undefined) ? \n null : this.items[this.valueToIndex(value)];\n this.$.selection.select(item);\n },\n\n updateSelectedItem: function() {\n this.selectedItem = this.selection;\n },\n\n selectedItemChanged: function() {\n if (this.selectedItem) {\n var t = this.selectedItem.templateInstance;\n this.selectedModel = t ? t.model : undefined;\n } else {\n this.selectedModel = null;\n }\n this.selectedIndex = this.selectedItem ? \n parseInt(this.valueToIndex(this.selected)) : -1;\n },\n\n valueToIndex: function(value) {\n // find an item with value == value and return it's index\n for (var i=0, items=this.items, c; (c=items[i]); i++) {\n if (this.valueForNode(c) == value) {\n return i;\n }\n }\n // if no item found, the value itself is probably the index\n return value;\n },\n\n valueForNode: function(node) {\n return node[this.valueattr] || node.getAttribute(this.valueattr);\n },\n\n // events fired from <core-selection> object\n selectionSelect: function(e, detail) {\n this.updateSelectedItem();\n if (detail.item) {\n this.applySelection(detail.item, detail.isSelected);\n }\n },\n\n applySelection: function(item, isSelected) {\n if (this.selectedClass) {\n item.classList.toggle(this.selectedClass, isSelected);\n }\n if (this.selectedProperty) {\n item[this.selectedProperty] = isSelected;\n }\n if (this.selectedAttribute && item.setAttribute) {\n if (isSelected) {\n item.setAttribute(this.selectedAttribute, '');\n } else {\n item.removeAttribute(this.selectedAttribute);\n }\n }\n },\n\n // event fired from host\n activateHandler: function(e) {\n if (!this.notap) {\n var i = this.findDistributedTarget(e.target, this.items);\n if (i >= 0) {\n var item = this.items[i];\n var s = this.valueForNode(item) || i;\n if (this.multi) {\n if (this.selected) {\n this.addRemoveSelected(s);\n } else {\n this.selected = [s];\n }\n } else {\n this.selected = s;\n }\n this.asyncFire('core-activate', {item: item});\n }\n }\n },\n\n addRemoveSelected: function(value) {\n var i = this.selected.indexOf(value);\n if (i >= 0) {\n this.selected.splice(i, 1);\n } else {\n this.selected.push(value);\n }\n this.valueToSelection(value);\n },\n\n findDistributedTarget: function(target, nodes) {\n // fi...
I guess you should create a bug report in the Angular.dart repository.
I filed an issue in the Angular.dart GitHub repo https://github.com/angular/angular.dart/issues/1227
