Are you sure you're using CrossOver's version of the "wine" command? We don't install ours in one of the usual PATH locations precisely so that we don't interfere with another installation of Wine.
Another reason to suspect you're not using our "wine" is that ours doesn't respect the WINEDEBUG variable. You are using it to suppress all Wine diagnostics output. Now, this is a bad idea when you're trying to debug a problem, but the point is it wouldn't actually work if you were running our "wine", and you'd almost certainly see a bunch of diagnostic output. (What you're seeing above is Steam's own diagnostic output, not Wine's.)
So, replace "wine" with ~/cxgames/bin/wine (or /opt/cxgames/bin/wine, depending on where you installed CrossOver Games). And leave out the WINEDEBUG=-all. If you want to control the debug output from our "wine", you can set CX_DEBUGMSG or pass the --debugmsg=<whatever> as an option to the command.
Edited to add: And CrossOver Games doesn't put its bottles into ~/.wine. So, this really seems like you're not using CrossOver Games at all. Have you tried using CrosOver's GUI tools to create and configure your bottles, install software, and then run it? This might avoid all of your confusion.