I put 3 but she is 4 years old and going to be 5 in the spring.
That's good, as a horse's spine doesn't fully 'close' until around 5-6yo so I don't like to do much ridden work before that. If you're light & doing light riding, I don't think it will be particularly problematic at her age. **Bareback riding is good even if the saddle fits well, and be conscious that the saddle should be checked for fit & adjusted/exchanged frequently on a growing horse.
I was told by the vet that I could start riding her on that day.
Yeah, by all means ride her, but I wouldn't use a bit. Perhaps the vet reckons the bit will sit far enough away from the injured area where the teeth were pulled, but there is likely to be generalised pain & maybe inflammation in that area, and you've noticed she was sore, so I'd take her word for it over his.
Also what is wrong with riding in the round pen that's what everyone says to do is ride a horse and work a horse in a round pen.
Oh riding in it? When I picture a roundpen I picture a smallish enclosure that people lunge their horse in. If you're riding in it for an hour, I presume it's a super-sized one - more like a small arena? If that's the case, not much wrong with that, aside from riding/training a greenbroke horse for a whole hour - short, fun sessions are better IMO - and the potential for boredom. If it is a regular size pen, nothing wrong with walking & a bit of slow trotting, but as Punks said, it's physically not good for them to do too much/hard on circles.