Correct or colloquial and grammatically wrong? My dictionary says got got the same meaning as have in american english and I have often used it instead. Would I make people correct me if I were to use this in a conversation? How bad does this sound to natives? I know "What have we got here?" is perfectly fine. Do people even use this in english speaking countries or is this foreign slang?
What do we got was probably not intended, but whaddya got or whadda we got, the compressed verbalization of what've you got or what've we got, in turn a contraction of what have you got or what have we got? Alternatively, it may be a mental conflation of what have we got and what do we have, which would be more widely accepted ways of asking the same thing.
Regardless of the explanation, it is the sort of slip in speech that would go unnoticed in everyday conversation, or in television scripts, especially because it is already casual. If whaddya got here? is meant to ask literally what is on hand, for instance if asking what flavors are available at an ice cream parlor, you could more formally say What do you have? If you are asking a subordinate for something that was assigned to them, you might ask What do you have for me? If it is an exclamation of surprise, you could ask What do we have here? But in conversational American English, these all sound rather stiff, and can be distancing.
I am less sure about British English, as have, have got, got and have gotten are all used differently— it is, some say, the most distinctive difference between the dialects— and so the world is full of opinions disparaging one or the other use as too informal or just wrong. You'll find notes on the matter in good dictionaries— see for example get in the Merriam-Webster Learners Dictionary or at Oxford Dictionaries Online, or various blogs.
Using "got" instead of "had," like in your example, would be considered slang. In fact, to most people, it would sound somewhat unintelligent. However, most people would also not correct the sentence, as it's pretty common and the meaning is still clear.
American English is weird at times...
"What do we got"
The meaning here is clear, but it sounds very clumsy. On the other hand...
"What have we got here"
Makes sense, and does not sound too terrible, at least in American English. Most people would not think twice about this sentence, although professionally, "What have we here" sounds much better.
To sum up the answer, you can use "got" instead of "have," but it does not sound entirely right, and actually sounds a bit unintelligent. Nobody would correct you, and everybody would understand your sentence, by typically, "have" in this scenario sounds better and would be more widely accepted.