Which bit-rate to use for gaming?

Most music is 16-44 so 44.1 KHz would be the most appropriate setting.

Games and such seem to all have audio at 48000 Hz, which is the default for Windows and most soundcards, not sure why (this is also what the digital stream output when I set it to auto).

The setting you mention is probably best. As long as you use WASAPI or ASIO in your media player, it will typically handle the bitrate switching on its own, and games will probably output at 48 KHz by default.

16 or 24 is easy.

If you set it to 16 all 24 bit audio (if you have some) will be truncated to 16.

If your audio device supports 24, set it to 24.

It will play 24 bit right and if you play 16 bit, 8 zero bits will be padded.

This won’t affect sound quality and might even be beneficial when using digital volume control.

You might wonder indeed if up-sampling e.g. CD audio (44.1 kHz) to 192 kHz improve things or degrade them.

Not all SRC (sample rate conversion) is without audible artefacts, not all DACs perform best at 192.

You might have a look at media players supporting WASAPI.

When done right, they will see to it that the audio is played at its native sample rate.

This of course requires hardware supporting this rate….

I have as many games in 44.1 as in 48.

Oddly I can notice the difference when my Xonar is resampling so I try to make it match when I play games (yeah, I can be bothered). Of course I pay attention to that with films (reclock+KS) and music (foobar+KS) too. 

I'm on XP and the Xonar can still resample when using Kernel Streaming (if the khz settings in the card's control panel doesn't match with the source), only ASIO can bypass it and with ASIO I can only have one sound stream while with KS I can for example play with the game sounds while still having my music work fine with no noticeable impact on the sound quality. 

To find out which games are running in 44.1 or 48 (a few of them mix both.. stupid devs) I use DPC latency checker, because when the Xonar is resampling the latency has an easily visible latency increase. 44.1 stream + 44.1 in Xonar = 45µs, 48 stream + 48 in Xonar = 45µs too. Same with 96 and 192khz (only got a couple of songs though). However 44.1 + 48 = 50-60µs, with 96 in Xonar it rises up to 70-80µs and 192 makes it skyrocket up to 110-120µs (!). I guess that's because the Xonar does everything using the CPU... 

Either way if you don't want to bother yourself too much with it I'd recommend 48khz/24bit as most games these days use that and all films (Blu Rays and DVDs) are in 48khz (either 24 or 16bit but it doesn't matter when you hardware supports 24+). I've heard of 96khz Blu Rays but haven't seen one, yet I do have a lot of Blu Rays. 

And yes WASAPI works fine for both music (many players to choose from, foobar heavily recommended) and films (you need Reclock for that I think, and to turn off the clock adjustment otherwise you don't get bit perfect).