I started lessons at 5 (my M. was a rider, but wanted a pro to teach me) and was competing by 8 with my M.. We've owned 5 horses at a time and competed on a weekly basis until my baby sister was born and I started having health problems a couple years later.
That school sounds HORRID. Don't switch to private there, switch to a different school!!!
Try a new school ASAP, you may not turn out to be horse people but that is NOT the environment to find out. It could also end up giving your daughter a long-time fear of horses or even animals in general.
Also, the horse constantly "stubbing" his toe could very well point to over-grown hooves, a common neglect in over-crowded or simply sub-par schools.
And horses who "attack" each other should not be used for lessons, period. Many advanced riders chose difficult horses that DO have dominance issues because they CAN and DO control it. I myself have ridden studs on a regular basis, who are often MUCH more dominant. If you have the skill, it's not a problem. BUT for lessons??? There's no excuse for repeat incidents of this during lessons.
The lessons should under control btw, children, novice riders and large, powerful animals are be a bad-match when things are not well-managed. Occasional (RARE) incidents DO happen in even the best of schools, but this school sounds like a poor excuse for lessons. Try a new one, look into it well, observe a lesson or too before starting them.
Also, I do find group lessons to be slow and boring just because of having more people learning at the same time. Private can be MUCH more fun and you can learn better and faster.
Good luck, have fun and be safe!