I'm also of the mind that tetras do better in something larger like a 15 or 20. I mean yes, they can do ok in a 10 gallon but there are other more suitable fish for a 10 gallon. Neons while small are very active fish and more space is best for them.
I would certainly not put the betta with the neons in a 10. With neons being so active there is a pretty good chance they are going to stress out the betta in a tank that size.
There is no solid yes or no answer on things that will be fine with a betta. It really comes down to it's personality. Some are good with tank mates and some won't even allow shrimp or snails. Most often they do best alone.
Things that will do well in a 10 gallon would be guppies, endlers, nano fish like chilli rasbora and celestial pearl danios.
If you get the tank fully cycled and are maintaining great water quality you could do the betta with a shoal of otos. Otos are sensitive fish so you need to be sure there is great water quality but they would most likely do well with the betta.
