Im a guy btw
You will start as a level 1 gymnast of course! Yes it is to late to be in the Olympics but why not do it for fun!?! Im a level 10 gymnast (very very rare) and I have been competing in the Olympics for 4 years now. TONS of gold medals!
15 is an okay age to start gymnastics if your doing it purely for fun. Most gymnasts will start at a very young age because they can build flexibility.
My friend is 15, almost sixteen, and is a level eight gymnast, and the people in her level, at her gym, vary in age from 10 to 16. To be an elite gymnast, which is not only difficult, but also costs a lot of money, you'd have to do a lot of training from a really young age, and the age range of elite gyms is extremely young, from what I know, going from 13-19.
It's difficult to say about wanting to be a gymnastics coach when she gets older. It would very much depend on where she wanted to get a job, and not so much as how long she'd been doing gymnastics, but her skill level. Some of my friends assistant coaches had never even done gymnastics as a kid, but were just fans of it.
It takes a lot of flexibility training, strength, and knowledge of how to hold your body. Floor routines are a lot of dancing and choreography with a lot of flips thrown in.
The people in television shows and movies are usually elite gymnasts, and look so graceful and natural doing what they do, but they have to work a lot harder, and they're generally doing extremely difficult stuff, for anyone to do. Gymnastics is a lot of basic flips even at level 8, with the occasional different kind of a flip.
It might be too late to take up gymnastics as a professional athlete (the pros spend their whole lives training in the sport), but I don't think it's too late to take it up as a hobby. Your body will still be in it's prime for a long time, and doing regular exercise might keep it that way for a bit longer time than the average person.
As for where to start, I would just look for places that teach gymnastics and take some beginner classes (thought it may or may not be hard to find beginners classes for people as old as you, I'm sure it's not impossible.)
Its not to late its gonna be harder tho cause you haven't had the years to train and make your body as flexible and strong as it could be.
Id go to a local gymnastics studio and take lessons with a trainer.
Its never to late! goodluck