Hi Bright I's welcome to Fish Lore
I'm sorry your having trouble with your fish.
Neons are pretty, but they can be pretty nippy, as you found out.
There could be a couple things going on that's making the older ones go after the newer ones.
Sometimes, if the fish feel crowded they can show signs of aggression. 11 neons in your size tank is kind of pushing it.
6 or 7 really is a good number.
You can try adding more plants and decorations to give the newcomers more places to hide and get away from each other.
If your water parameters are off this can also effect how your fish act.
How old is your tank? I see by your aquarium info that you don't know about the
nitrogen cycle.
Are you testing your water? If so, what are your readings for
ammonia, nitrites and nitrates?
Here's good article to read about the cycle.
http://www.fishlore.com/NitrogenCycle.htm
Your neons tails should heal fairly quickly with nice clean healthy water to prevent infection.
Good luck, and sorry about all the questions, the answers will help the members help you with your tank.