welcome to gymnastics!
I hope that this helps a little: (i also have a 5 year old gymnast)
the 5 yo...strong, catches on quickly, only wants to do gymnastics...Sounds like gym is the place for her to be right now. At 5 years old, gymnastics should be fun. As far as "What should a 5 yo be able to do" - honestly? My answer has to be: stay in class, pay attention to the coach, and have a great time learning gymnastics. At 5 years old, there really isn't anything specific that a child should be able to do. Skills aren't age-specific. For instance, my 5 year old is training for level 4 and level 5 currently, but there are SEVERAL other 5 year olds in our gym who are learning front and back rolls, cartwheels and handstands and bridges.
How do you know if you are in the right gym?
If your dd likes being there, likes the coaches, likes the kids, is having fun and learning at her own pace, then I would say you are probably at the right gym for you.
How do you know if she is being pushed too hard/not hard enough?
I'd say...since she's 5, gymnastics should be fun. So as long as she is having fun and progressing at her own pace and not being forced into doing things that she doesn't want to do as a 5 year old, then she's not being pushed too hard. If the coaches are spending more time standing around and talking rather than teaching their classes, your daughter is bored and isn't having fun and learning, then maybe she needs a class with a little more structure and challenge.
What level do kids start competing and what age:
That depends on your gym and what they do. Some gyms start competing level 2 unsanctioned competitions. As far as sanctioned meets, some gyms don't compete until level 4, some gyms it's level 5. But whichever your gym does, it's really up to the gymnast (child) whether or not they want to compete. Being on team is a huge commitment for the child as well as the parents: time, financial, emotionally, physically, etc...All I know is that the minimum age requirement for USAG Level 4 is 6 years old. If the gymnast isn't physically or emotionally ready at 6 years old, then they won't compete and shouldn't. I know I keep saying it, but gymnastics should be FUN for kids and not something that pushes them beyond their mental capacity.
With your littlest one (3 year old) Is it better to move by skill or age?
By both, really. If her skills are a little more advance than the rest of the kids in her class, then there's really no problem in putting her in a class with 4 year olds. I think it really depends on the personality of the child. If she's cool with being in a class with girls a little bigger than her, then it should be okay. At three and 4 years old, gymnastics is somewhere that kids can get with other kids and have fun while learning about what their bodies are capable of (running jumping hoping rolling climbing swining...)