I have heard people say "Do you got a moment?" but it is not proper English. I would definitely avoid saying "Do you got a moment?" in a professional environment, but it is acceptable in a casual setting. Hope this helps :)

As you say, the two correct forms are 'Have you got an moment?' and 'Do you have a moment?'. There is no difference between them other than the fact that the first is more informal. 'Do you got?' is grammatically incorrect. 'Do' has to be followed by the base form of the verb, not a past participle. I believe it exists as a non-standard form in the US, and possibly in some creole dialects, but it is, as @Certainly says, most definitely 'not proper English'. If you were to say this to a British English speaker it would be assumed that you were a foreign learner with a very poor grasp of basic English. And no, we wouldn't say 'Do you get a moment?'. Why? It's grammatically correct, but it's not at all clear what it means. It certainly doesn't mean the same as 'Have you got a moment?'. As Andi says, it sounds like a fragment of a longer question about a general fact or a regular habit, for example 'Do you get a moment to talk to your friends between lessons?'.

I have never heard "do you got a moment?"and this is poor English. It would be correct to say "do you have a moment?" As for "do you get a moment?", this is grammatically correct but is a fragment that is rarely used alone, and always in the context of the workplace or when another person influences our time. For example, "do you get a moment to relax?, or "do you get a moment to have a break?"