It sounds like you're using common sense on this. I definitely would be very careful, especially with dips. If one ring breaks, you will be in a very precarious position. You cannot support yourself with only one ring.
The grip tape you see is a Power Rings thing only. I designed them from the ground up to be used for training purposes. Traditional rings are wooden and most gymnasts wear special grips and lots of chalk. I think the grip tape on the Power Rings helps for people that do not necessarily want to chalk up. For gymnastics though, the tape comes undone pretty easily if you swing and wearing grips will simply rip it to shreds.
The new Power Rings are made of steel tubing, 1.25" thick and extremely strong. They are powder coated with a light "crinkle" texture, so the grip tape might not be necessary. They are hung by 4000 lbs nylon webbing and a 4000 lbs. buckle. It adjusts quickly from overhead for pullups to the floor for pushups. They are not really geared towards gymnasts, but I have sold a few to gymnasts and clubs.
Daniel, I would be very careful with muscle-ups too as they can be somewhat ballistic, especially on the lowering phase. Same for crosses. Best of luck and be safe!
And good news on the newly redesigned steel Power Rings. I am expecting that the first ones will go out the door today (friday) and all orders will be fulfilled by mid next week.