Are they new, if so they could be setting a pecking order, it will settle a bit but they will still be very active, my Gold Barbs finally settled down but in the beginning they were pretty frantic, now they swim calmly together because who the top member of there school has been determined, it would be different if I added new members to the school as well when you have an established school its sometimes very difficult to add new members, and the newbies usually end up by themselves
Sounds pretty normal Tetras tend to nip and pick at each other a bit when setting up territories then they also do that from time to time almost like they're playing, which they may be...but I don't speak fish so I don't know for sure
Do tetras have a specific season they mate in? I have noticed mine have been chasing each other around a lot and not schooling together so much. They were fine last week (they've been in the tank for 4-6 weeks now).
Fish will always do that, no matter how peaceful. I was so worried about my platies a while ago because all they did was chase each other, but I learned that it is part of life, they establish a pecking order. I actually tried to stop the chasing by moving the most aggressive to a small tank. It did not help, it even depressed him(little to no movement, acted very skittish), so I put him back. Just expect chasing from basically every fish, and only start to worry if you see signs of harm.
The Neons were in a 33g long (4' long) with Blue Kerri Tetras, Cory Cats and 3 very small (dime size) Angel Fish. I know for a fact the damage was not done by other tank mates. I saw them nipping each other with my own . It was sad to watch and I hate that I lost fish.