"have" vs."have got" in American and British English

On usage, the Cambridge Grammar of English (p883) states:

The present tense form of have with got used for possession is more than twice as frequent in spoken BrE as in AmE:

I've got one sister and one brother. (BrE)

I have a cousin who never married. (AmE)

On formality, Swan in Practical English Usage (p230) states:

Got forms are especially common in an informal style. ... In very informal American speech, people may drop 've before got. I('ve) got a problem.

This not a matter of what's more formal or more proper or more anything. This is a very complex subject, with a number of constructions, and rules, and idioms, and complications involved. The presenting question covers one small tip of the iceberg; this answer covers the iceberg. Oh, and this discussion is about American English; I take no responsibility for UK usages.

First, the verb get has two basic meanings, both grammatical rather than lexical
(in the following, "causative/inchoative" means 'come to be/become/cause to become'):

get can be the causative/inchoative of the auxiliary verb have, in all its uses
She has/She got leprosy. She has/She got a car. I had it done/I got it done.
= She came to have leprosy. She came to have a new car. I came to have it done.
get can also be the causative/inchoative of the auxiliary verb be, in all its uses
She was/got married by a priest. He was/got going fast on the hill. They were/got tired.
= She came to be/became married. He came to be going fast. They came to be tired.

Second, the present perfect construction uses the auxiliary have, and occurs in its Stative/Resultative Perfect sense with 'come to have' get ; in two variants: have got and have gotten. We only deal with American usage of have got here.

Third, since get means come to have, and since this is Stative/Resultative perfect, has got means 'has come to have'. OK, except that if one has come to have something in this sense of the Perfect, it must be true in the present that one still has it. So

I've got a car = I have come to have a car = I have acquired a car = I have a car.

which immediately confuses the have of the perfect with the have of possession.

Fourth, auxiliary verbs like be and have are almost always contracted (they're auxiliaries and have no meaning; they're particles, intended to direct your attention, not to hold it). This means they're reduced to final consonants cliticized to subject NPs, mostly pronouns.

I've, I'm, I'd, You've, You're, You'd, It's, It'd, We're, We've, We'd, They've, They're, They'd

Fifth, these clitics in turn get deleted whenever possible. After all, we already can't tell the difference between the contraction she is and she has, or between we would and we had, and that doesn't bother us. Plus, English does not prefer initial clusters with /ɡ/, like /zɡ vɡ dɡ/, so /z v d/ are lost most of the time before got.

You've got the answer -> You got the answer.
I've got the answer -> I got the answer.
He's got the answer -> He got the answer (reified by children to He gots the answer.)

Finally, since have is more often used as an auxiliary than to mean possession, and since the have got construction means the possession sense of have, it has come to take over the possession sense of have.

This is syntax in action, still changing, different now from the way it was 50 years ago, and coming to have new senses and new uses and new distinctions every day. It's alive, after all.

To my BrE ears, ‘I have got a cat’ sounds like a full sentence, whereas ‘I have a cat’ sounds like a lead-in to a fact about that cat, such as if it were followed with, ‘who chases her own tail!’ It also sounds like the person has several cats, but is only talking about one of them.

Do bear in mind most people will use the contraction I've got in speech, and it does sound more informal than I have.

You are correct about the negative and interrogative too: