^^I strongly agree with Fabulous that there is no real set age^^
My dd started "late" too. She was 6 1/2 when she started gym. She did 6 months of rec and a year of pre-team, where she competed a few mock meets at L3-L4. While I can't say I knew how far she would go, I can say that she picked up skills very quickly at that time, and she seemed very polished compared to others at that age especially based on her experience. She was strong, flexible, and didn't seem to have much fear.
At 8 (turning 9) she competed (old)L5, then L7 at 9-10, L8 at 10-11, L9 at 11-12, and competed L9/L10 at 12-13. She was not at the "best" gym in the area, but they certainly gave her a good enough base to move. We did ultimately switch gyms before L9, when I felt that the old gym could no longer support her progress. But it was not a decision I felt needed to be made until maybe L7-L8 because her former gym had a very decent compulsory and lower optional program, and until I was relatively sure that she would stick with it, which at your dd's age is not really something you can say for sure.
My criteria has always been that she is 1) emotionally healthy, 2) physically strong, 3) progressing constantly (not necessarily "getting" new skills, but working towards them), 4) that she feels challenged and 5) whether there is still room to grow.
Also, I have some experience in this arena with my younger DD so I can safely say that Yes, horses DO wear socks. They prefer mismatched ones, occasionally in colorful patterns!