Yes, you can do all this with the free, open source software - I am a Linux user and a Flash / Flex developer, and I am living proof !; -)
I write all the AS3 and MXML code in Vim and compile with mxmlc or amxmlc. There are some syntax definitions and omnicomplete plugins for ActionScript that work very well - not full autocompletion, as you can find in many IDE graphical environments, but enough to make Vim a very good option for developing Flash and Flex.
For the graphical user interface, you can create MXML layouts using designview AIR, a basic design alternative in Flex Builder.
I also have a short wiki article that may be useful for command line tools for creating AIR applications here .
If you are on a Windows platform, FlashDevelop is free and open, which is very good. I use .NET 2.0, but it doesn’t work very well with other OSs.
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