What is the best cross-platform game engine, being the most commercially supported, there is for free available for development on Linux? I'm planning on creating multiple, non-commercial, fps games to gain real industry experience in C++ game development.
Firstly, "best" is a rather subjective term, personally, I enjoy working with the Source engine (at least, the client and server which plug in to the base source engine, and determine your game play), making a MOD with a recent industry-standard engine is probably the closest you'll get to working within the confines of a restricted codebase in the industry. Others swear by the Quake engine for it's open-source qualities, and other's prefer various other engines.
If you are looking for industry experience, then I would say that it would be extremely advisable to develop your own engine, to get an idea of the scale of development that goes into producing a marketable industry-quality game. You will learn how to handle graphics development, AI development, handling performance and optimization of your code, understanding what goes into making a game cross-platform, and hopefully, have an enjoyable time. If you do decide to go down this route, I would recommend OpenGL for it's cross-platform ability, however, most games are written for Windows, so DirectX with an OpenGL fallback is also a possibility (this also allows you to learn about dynamic linking of DLLs etc.)
Although you may only be interested primarily in one field (AI Programming, Weapons programming, Physics programming etc.), building your own game-engine demonstrates to a potential employer that you are, firstly, passionate about game programming and dedicated enough to do something about it, and secondly, have a well-rounded understanding of game development, which is extremely desirable especially for a senior programmer).
Personally, other than the Source engine, I've never used another's engine for an FPS, however, I have heard good things about the cafu engine, which is supposed to be cross-platform and relatively recent (and with good access to it's rendering process etc.)
I hope this helps you to make a decision, and good luck with your future career in the gaming industry! :)
EDIT: With regards to your edited question, I believe OGRE has a lot of commercial support, and has a regularly updated code-base (and fulfills all of your other criterion). However, I would still recommend writing your own engine, for the reasons highlighted above. :)