Great ideas, thanks!
Oh dear, I do feel for him and for you. I've always reminded my kids that a gymnastics meet is busy, there are competitors on more than one apparatus at the same time - the only people that are probably actually watching my kids are myself, dh, grandma, the coach, and the judgesfor that apparatus, not everyone in the room is actually watching them. We talk a lot in meet season about how nerves are normal, everyone gets nervous before a meet and if he can conquer those nervous, anxious feeling then he will feel great afterwards.
One of my kids coaches taught them to acknowledge the feelings and to use them: I have butterflies in my stomach - they will help me leap higher, my heart is pounding really hard - that means I can run faster down the vault strip ... so trying to turn that jittery feeling into something positive.
Is the coach aware? I would have a quiet word in their ear if possible. I have one who gets really nervous, but is ok once we actually arrive at the meet, the other it doesn't bother as much - sometimes I think I am more nervous lol.
I have tried reminding him that the vast majority of people there are not watching him, as they will be too busy trying to see their own kids/teammates. I hadn't thought to teach him that the nerves can be a good thing, so I'll try that too.
Yes, I have talked with the coach, as DS is very good at hiding his feelings from everyone but his family. Coach understands and is very laid back. He puts no pressure on the kids to get a certain score, but sometimes I think he forgets just how young DS is. This was a big issue for us this year, because I felt DS should have repeated a level due to his age and his confidence issues, but the coaches felt he should be moved up. In the end, I deferred to the coaches, but I believe he would be less nervous if he were competing at a level where he was more confident with the skills. It's too late to go back now, though.
skschlag, that's funny that you mention dancing. I showed him a video of one of Lloimincia Hall's routines, and he was fascinated. He said he wished the boys could dance like that.

I told him instead to imagine that the judges were dancing like that, and he thought that was hilarious.
As for bringing a stuffed animal... he
does love his stuffed animals. He sleeps with a million of them, and definitely cuddles them as a comfort thing. I think that would work, except for the fact that he is the youngest on team. I think he'd worry about what the 13 year old would think and will not want to be seen as a baby.
I'll also talk with him about how important it is to cheer on his teammates. Last year he was in a smaller, younger group, and I don't think any of them really paid attention except when it was their turn. It was almost like they needed to zone out for a while in order to focus again for the next event. I think if he views it as more of a team sport where they cheer each other on, it will be more fun.