In the ethernet board, pins 10-13 are reserved. The lcd-shield is using pins 4-10 (although #10 is the backlit control, so it may be possible to go without it, just having a constant backlit) - I have the lcd - keypad shield, but at home, so I cannot check that now. - look into the lcd-library to see if you can somehow define it.
That is, it is not possible to use the backlight and the ethernet at the same time. For your other question, you just stack the LCD -keypad shield on top of your arduino board, no other connections or cables are needed.
Although it may look a bit more scary, a display without a shield, eg http://dx.com/p/16-x-2-character-lcd-display-module-with-blue-backlight-121356 is not much harder to set up and it gives you more flexibility. (although, for the display I linked to, you must be able to solder in a set of pins to connect it) Another alternative is http://dx.com/p/16-x-2-character-lcd-display-module-with-blue-backlight-121356 - the latter gives you more flexibility, but poorer readability and it is a bit more work to set up the library.
