Been trying to get games going in ubuntu myself and have had best success with playonlinux (pol) to launch steam; which on installation tells you "do not run programs installed by pol from default location" and although installing pol many times have only just recently worked out what it means rflol How are you launching "steam"?
What graphics driver are you using? Trying different drivers might give a surprise result just try and see if you want and got time. Just keep on using the same method for installing/uninstalling and uninstall any proprietary being used before trying another. Dont mean to state the obvious but adding debs, ppas and running .runs is in the long run unfruitful... Plenty info on web but have just come to the conclusion that using software center and/or additional drivers best thing if you want to play games and use a fully working unity/os.imho
Updating to the kernel you are using and installing 310 is what has worked best for me to get games running but have not spent much time running them lately... or sadly using that machine. It was a jaw dropping moment when first being presented with desktop using that driver. night and day.
Have you tried selecting virtual desktop in wine.cfg? This should help you get game running but might have to adjust sizes of desktop when activating virtual desktop to suit the size of screen you are using as the default can be a bit small.
Have you tried searching appdb at winehq for test results with the game you use?
Running anything through wine seems to have its own querks and different querks on different hardware. Check the comments in different test results to see what worked (and what did not) for your game at appdb.