I'm a mom of an almost 7 year who is not competing yet. But I see her get mad at herself when she can't get something. I'm suspecting it might get worse down the road.
Obviously I don't know you so I can't really answer your questions directly. But some observations - you can judge how close to home they are. And, they are going to be a bit all over the place. I apologize for that. Also, these are coming from my laymen's observation of people over the course of my lifetime.
For one thing, I'm not sure all of what you are talking about it is about being competitive with others. I think some of it is about you actually being competitive with yourself. I suspect that you might be a personality that strives for perfection in all that you do - not just gymnastics. And, when you can't accomplish what you want in any activity, I'm guessing that you are pretty hard on yourself.
Someone I work with often tells me "Don't beat yourself up because there are plenty of people out there who will do it for you." I actually think that is much better advice for a person who strives for perfection, which is just not attainable, then to remind the person that people are not perfect. As someone who strives for perfection, you just might be much harder on yourself than some of your team mates, and you may not see your true abilities, because you are busy kicking yourself. If a coach compliments you and you have a hard time believing the coach because you think the coach is just being nice, you just might be being too hard on yourself. Without being arrogant or otherwise feeling better than others, it is not bad to take a long hard look at what it is you can do and do well and be proud of those things and try to give yourself some time (while working hard) on the other things.
Along with being a perfectionist, keep in mind that your body is something that you can control only to a point and there are just things beyond your control. For instance, developing certain muscles is a process that takes certain repetition and time. There is no way that the human body can be made to build muscle overnight even if a person desperately wants to. The same must apply to flexibility and other essentials for gymnastics. So, when you are working on something new, it is not all sheer willpower, but it is being patient with your body to catch up with your mind.
As for your team mates who you think might not care when they can not accomplish something or if they fall at an event, a few different things could explain that too. One might actually be that a team mate is really just the type of personality that wants to have fun and winning is not a necessity. But there may also be those team mates who really truly are upset about not accomplishing something or falling in a competition but who internalize (hide) those feelings. Such a person might also be a perfectionist personality who can't or won't show his/her feelings to others. There could also be other reasons that you don't know for why a person may not seem to you to care.
Ok, now I may be getting way too philosophical - too late at night for me, I guess.
Take care and don't beat yourself up.