Your problem might just be that trackmania may not keep everything in it's folder (I can't check right now) and so crash due to missing files. So install the game under wine (it works) and your problem may be solved just by that. Read the appdb to add the openal DLL linked on the appdb page. But don't install DirectX as written in the appdb, just add the missing DLL (there is a thread about this on this forum)
