A:Answer Well for starters the Ultra can also be used on 4k tv's, the regular one cannot. But if you don't have a 4k tv, I still say get the Ultra anyways because it also has a superior Wi-Fi signal than the regular Chromecast, that means less wifi interference and shorter casting times to your tv. Plus if you do get a 4k tv in the future, your already set.
I haven't tried playing games from phone to the tv, but it should be really easy. All you have to do is "mirror device" to the tv using the Google Home app. It'll mirror anything from your phone screen to your tv screen.
I also recommend buying an HDMI extender to put some distance between your Chromecast and TV.