If the MSDN AA is similar to other MSDN subscriptions, the keys will still be valid after the subscription expires. MSDN AA likely differs from an Ultimate subscription, though- so as others have said, check the EULA. In this case, it would be in the MSDN Terms of Service.
As well as the Visual Studio .Net 2002 had an annoying blat on compiled programs stating that its the "Student Edition" and "Not to be used for commercial use", and so other than tinkering with it and making programs for myself and ignore the annoying blat message, I couldnt really continue use with it past college because why have to work in this and then export the source and compile with a different fully paid version of Visual Studio .Net ... you pretty much need to buy the full blown official license if your going to continue to use it beyond college for commercial use. You might find the same problems with the software you have or get that it has a watermark or some sort of blat that gets tagged to projects and cant get rid of unless you buy the full blown official commercial use package of the software.
Microsoft has provided the .NET Framework SDK for free since their first releases, which, coincidentally was used with "Visual Studio .NET" released in 2002. Visual Studio .NET itself had Academic, Professional, Enterprise Developer, and Enterprise Architect Editions. I would suspect making an academic version work without such a prompt at the start would be a case of compiling the program manually via the command-line compilers from the free .NET SDK. Thankfully they wisened up and now provide Express Editions of Visual Studio.