@rjack321 did a great job addressing these, some additional observations:
If so, is it not possible to take the mouthpiece out of your mouth and breathe air?
On CCR you are encouraged to stay on the loop wherever possible on the surface, especially post dive as it helps with deco (basically breathing 70 percent Oxygen on mine, I will stay on the loop until I can't take the odd looks on the boat anymore after a tech dive. My profile pic, in fact, is about 30 min after surfacing from a 60m dive)
was there no air in his wings?
CCR's require discipline in a lot of ways, one of them is closing the loop before removing it from your mouth. Passing out makes that impossible, so in LOC (loss of consciousness) events on CCR, they are very often accompanied by a flooded loop and a suddenly VERY heavy unit. I have seen several divers using wings too small to support them and a flooded CCR, add in all the deco tanks and chronic overweighting of tech divers at the start of a dive and things can get really difficult really fast.
WHY do instructors let students into the water with cameras?
Balancing "nice guy" vs instructor, "fun dive" vs "training dive", especially in a place where you are likely to see very cool stuff on the training dive and it starts getting easier to normalise the deviance of taking cameras on OW training dives. I only started taking a camera with me on CCR dives after I hit 100 hours, I did not feel comfortable with the added task loading before then. Others feel differently. I will only take a camera with on a training dive if it is on a mask mount and I start before I splash and turn off when I am back on the boat.
Do tec divers not do buddy checks or was it just missed?
Tech training is very big on pre-dive checks of the team, and generally tech divers are less likely to skip checks, but everyone is different and it only takes one bad day.
Most of these rules are written in blood, not ink, sadly it sometimes takes a bit of blood to touch up the writing so everyone can see it clearly again. I hate that it works this way but it is what it is.