Flange hole-- about 3 or 4 inches... However the main reason I asked is in case the existing flange was considerably deeper. Yours is almost flush, so a Push-tite flange won't be needed.
I am not a plumber or other pro. For the toilet purpose, I think the repair flange would be good at the job.
Gap around the outside.... hmmm. What is under there? I would like to see that filled with something that would keep bugs from traveling and would block air flow if that area was a crawl space. I am uncomfortable with that, and the pros on the thread are really not liking it.
A repair ring will secure the closet bolts. The repair ring itself is screwed to wood outside the main circle, although that big hole in your wood does look excessive. That could be a problem. There is another type of split repair ring that I don't think would be good for this at all. This thread shows a picture:
https://terrylove.com/forums/index.php?threads/where-to-buy-red-ring-flange-repair.5267/
From the standpoint of sealing the toilet to the flange, wax will do a good job.