Welcom Flipforme#1. Hopefully someone, a coach, can reply to your question regarding the back tuck.
My daughter has been in gymnastics now for about four years since she started on the recreational side. She is now 11 and in level eight. Whether it is the sport or something in their own character, gymnasts tend to be perfectionists and demand a lot of themselves. The sport itself is very difficult and the gymnast does not need anymore pressure from a parent. So don't be too obsessive. Reign yourself in a bit. I try to be emotionally supportive of my dd and encourage her to be the best she can be and just have fun. I rarely ask her in detail about the different skills she has learned or is trying to learn. I generally ask if she had a nice practive, anything new... Of course if she wants to volunteer something, I listen.
If your daughter does not enjoy gymnastics, she won't be willing to put in the effort to acquire new skills and get better. If she can combine the hard work in the gym with friendship with her teammates, having fun, etc, she will be less likely to complain about any diffiiculties or even be conscious of all the hard work she'll need to do to improve her skills. Don't necessarily expect her to be great. Even if she doesn't become a great gymnast, she can through gymnastics gain valuable qualities such as determination, resilience, focus, patience, etc, etc. This will, in the long run, make her a big winner.
But you also must train like your daughter. To be a good gym parent, the skills you need to master are patience, restraint, discretion and so on. Be supportive, encouraging, etc. Don't push too much. Don't put pressure on her. Don't let her see you ....sweat.