The whole point of AIR is to enable web developers who know AS/JS to be able to easily make desktop applications which are automatically portable to win, linux, and mac. Hence why there is no way to connect it to other languages like C#, C++, etc, since web developers generally don't care about those languages, and if they were used, the app is no longer portable (yes, most languages can be ported to other OS's with some effort, but the point of AIR is that you write whatever you want, and it runs everywhere with no extra work).
If you want to combine C# and Flash, instead of trying to put C# inside AIR, what you should be doing is putting Flash inside a C# app. There are various resources for this on the internets, here is one I found in a quick search: Example: Using the External API with an ActiveX container. When you are searching, make sure it talks about the newer, more advanced ExternalInterface / External API method, and not the oldschool SetVariable/CallFunction/watch method. The newer one is much easier to use & has more features.
Also, yes, you can use custom Flash components inside a Flex app. Generally you just make a SWC inside Flash CS3 (or CS4), and then import that SWC inside Flex Builder (or Flex SDK, or FlashDevelop, or ...)
If you are making the components yourself inside Flash CS3, you should use the Flex Component Kit. It provides some features for you, and puts in the boilerplate code, to make your flash component play nice inside Flex. It's included with Flex Builder/Flex SDK.