If I do a speedrun on my Xbox Series X using backwards compatibility with an Xbox360 disk, do I submit the run as played on Xbox360 or Xbox One?

Probably Xbox Series X because it's the hardware on which you play that matters. A mod will tell you soon enough

I've added series X/S.

I can use a livesplit overlay on my speedrun through OBS. I don't believe the autosplitter supports console. Do I need to have an autosplitter in my console run?

you do not

The autosplitter is simply a tool to relieve the runner of having to manually split, the final time is determined by hands from the footage and the loads in it.