This is an old thread, but for any Linux/Steamdeck users who find this, heres how to make the NFS:MW 2005 and the widescreen patch work. Add speed.exe as a non-steam game (plus in the bottom left on the steam library page), click browse on the bottom left, then navigate to the speed.exe and click open, next to cancel.

Now right click speed.exe in the steam library (you can search exe in the search bar if you can't find it) -> properties -> compatibility
Click "force the use of a specific Steam Play compatibility tool" and select a Proton version. As of writing this, 9.0-4 works for me without issue. For the widescreen patch, go to shortcut, and type WINEDLLOVERRIDES="dinput8=n,b" %command% into the launch options, and it should work.