When American born, bred, raised, etc. singer-songwriter Edward Joseph Mahoney sings
I've got two tickets to paradise (youtube link)
(Compare: I've got two tickets in my pocket)
he means:
I possess two tickets to paradise
not
I have acquired two tickets to paradise
which in AmE would usually be expressed as
I've gotten two tickets to paradise. (AmE present perfect)
whereas in BrE it would usually be expressed as
I've got two tickets to paradise. (BrE present perfect and 'double form of have')
Typical BrE usage of I've got as present perfect:
I've got Pippet a new jumper, think she'll like it?
(In AmE, we would say I've gotten Pippet a new sweater.)
Now you should know that I've got can mean I possess in both BrE and AmE.
The Beatles provide an example of BrE in I've got a feeling, while another two of many many American uses are those of Frank Sinatra in I've got you under my skin and Garth Brook's (I've got) friends in low places (1:13 etc).
Note that in both BrE and AmE, I've got can also express obligation, as in
I've got to clean my room before I can go to the concert.
"Between you and me" is historically correct, but many native English speakers -- perhaps especially my fellow Americans -- do not know this. I suspect that the confusion comes from the fact that "you" and "you" are indistinguishable, so people get used to "You and I" as a subject and then use "you and I" as an object. This is not helped by the fact that there is much emphasis placed on correcting "you and me" (and "me and you") as a subject into "you and I".
The disagreement I think comes mostly from descriptivists who say that whatever is understood and commonly used becomes correct.
In addition, song lyrics push the limits of acceptable grammar often, sometimes on purpose and sometimes just because -- as in your example -- it makes a better rhyme. We have an idiom of "poetic license" which is used to excuse grammatical as well as factual inaccuracies in the name of art. See also for instance "Say a little prayer for I" in which the object (properly "me") is changed to "I" for a rhyme even in the absence of the word "you".
Best Answer
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.