Partially answering my own question

I had a play around with the Q10 over the weekend. Plugging in an Xbox 360 game pad doesn't work; it receives power, but is not recognised by games/emulators. Same result with using the wireless 360 adapter.
I then rooted the Q10 using the LAN method posted on this forum - painless and very quick to do. I followed the guides about copying specific key layout files for both wireless/wired 360 gamepads, but unfortunately got the same result. It would be nice if HiMedia could think about adding native support for this at some point, like the Nvidea Shield TV for example.
I also tried to pair a DualShock 4 with the Q10 (the PS4's controller). It pairs fine with my Android phone, but hangs at the pairing stage with the Q10 so also can't be used in this way - people do report patchy compatibility though with various devices.
Finally, I tried the Sixaxis controller app - this worked with both the Dualshock 3 (PS3) and Dualshock 4 (PS4) controllers, so at least I have a solution. Note this also requires root if you want to connect wirelessly.