No, we don't use 'revenge' as a verb. This is not an issue of formality or regional differences, either. No native speaker who uses English correctly would normally use this word as anything other than a noun. If you come across a supposedly correct use of 'revenge' as a verb, it could possibly be a very archaic usage - in which case you should ignore it. None of your examples are possible, I'm afraid. The only one where you could replace 'revenge' with 'avenge' is the third. With the others, we would probably use a phrase such as 'seek/take revenge' in more formal language or 'get (his) revenge' in more informal language.

"Revenge" as a verb doesn't sound natural or right to me at all. I would say something like "He got revenge against his enemy" or "He got revenge for his friend's death".

No, it's not a verb. You say 'take revenge on', 'get (your) revenge on' or other variations on that theme. 'Seek revenge for' or 'want revenge' are other possibilities. The only one where you could possibly use 'avenge' is the third sentence. The context in which you use 'avenge' is much more specific. If you avenge the death of someone we normally take that as meaning you kill the person who has wronged you. Or perhaps doing the same thing to them that they did to the person you care about. But normally it has to do with death.

Thinking about it, it might be used as a verb, but really only as a reflexive. E.g. 'he revenged himself against John' meaning 'he got revenge against John'

I would say no.You might take/get/extract revenge. or you could avenge something.