Why in the world is it trying to open Adobe Flash Professional when building or running the project?
So I am trying to build my Flash project
I am given this, I don't have a license to Adobe Flash Professional anymore, so I can't use this software.
What can I do?
FlashDevelop is NOT a full "alternative" to Adobe Flash Professional nor Flash Builder.
It is an IDE oriented to supporting the AS3 language and also Haxe, HTML, XML, Javascript... But as far as I've used it, it doesn't have facilities to manage a timeline or library in a FLA file.
For AS3 development, FlashDevelop is great and way better than Flash Professional. It's more like Flash Builder in that sense (although Flash Builder is more oriented to Flex/AIR and has facilities for creating UIs). I used earlier versions to edit/refactor the AS3 code associated to an FLA in order to build SWFs for Flash Player, and in this way it relies on Flash Professional to create the final output.
However, FlashDevelop also uses the freely available Flex and AIR SDKs to build applications, and you can go a long way if you're willing to handle the images/audio/video resources yourself in AS3 code. Select the appropriate option when creating a new project, it will create simple skeleton applications for you and they should build with the Flex and AIR SDKs it downloads during installation.