Welcome to the forums and thanks for your feedback, appreciate it very much!
The game list is currently pulled from moonlight and it shouldn't be altered in any way. You could see if it is indeed listed twice using moonlight's CLI:
That being said, I'm currently working on using HTTP requests directly to the host machine and thus getting the information directly from GeForce Experience. This obviously changes the current behaviour and allows me to get way more information about the games installed and it might also fix your problem (and if it doesn't it means that a single game is being reported twice by GFE).
This might very well be the case - any chance you could set it to 'always allow' or something so the popup doesn't appear every time?
That's somewhat strange ... mostly because it's working in the settings / menu but not in the game itself. You can try passing the relevant evdev device to moonlight as an input source. You should probably do this using moonlight's CLI as well because Luna's latest released version doesn't support multiple input sources right now (this feature is already finished but I need to finish some other stuff as well for the next version):
shutdown Kodi first:
sudo systemctl stop mediacenter
moonlight stream [host] -app ["Application Title"] -input [device1] -input [device2]
(if you don't know which event corresponds to your X360 controller or keyboard, you can list that information using
cat /proc/bus/input/devices ... or just try launching moonlight without any specific input option, it should use all of them in this case)
after testing, relaunch Kodi using:
sudo systemctl start mediacenter
Spaces in game titles have been fixed for a loooong time, only the first few versions had some quirks when it came to this. However, GFE doesn't support all games which means it doesn't list them automatically - if you want them to be listed you need to add them manually using GFE's settings ('Settings' on the top -> 'SHIELD' on the left -> there's a big grey box where you can add game locations).
Let me know if this already helps. Most of this stuff is moonlight / GFE related though and happens at a point where Luna is not even running anymore, so I can only try and support you as good as I can (and obviously try to fix things if they really boil down to being an issue with Luna, as your first two points might be).