I have two questions:

1. There are currently no runs posted for a lot of the categories. Does that mean if I submit one that I would have the record by default, or am I missing something?
2. The rules say "Timing starts upon gaining control of Link in the 1st room of the dungeon" but the runs that are listed all seem to start the timer during the sequences outside of the divine beasts where Link fights his way in. Are those sequences considered "rooms" in the dungeon?

Looks to me like the timer starts upon control when fighting your way into the Divine Beast. Makes sense cause that is an intrinsic part of the Divine Beast. When all else fails follow the lead of already approved runs. Also it's best to start the timer too early than too late, though you don't even really need a timer.

In terms of submitting a run to a category with no other runs, yes you will technically have the world record for that level according to speedrun.com but this site doesn't encapsulate anything. Also, just because you finish a run, if it's a poor run you shouldn't be content with it. World record or not, if a time can be improved your goal should be to improve it, though that's also dependent on the value you put on speedrunning.