They are the same. There is no pattern to their use, except I believe in the UK they prefer the "have got" version. We use both the US and they sound perfectly natural.

"Have you got" is more common in British English than "Do you have" ; but "Do you have" is much more common in American English than "Have you got" The word got generally bears a more informal sound to it, maybe because of its wide range of uses. In contrast, the word do, when used in this manner creates a structure that is probably unique to English. The latter option displays the verb-subject word order used in many other languages, while the former uses the do+subject+verb construction, which probably appeals to the native speaker's ear.

They both have the same meaning, but I can't imagine using "have you got...?" in a formal situation. It's worth knowing both instead of using only one.

They are both correct and both have the same meaning. The only difference is that 'Have you got?' is more casual and informal. It's very widely used in everyday speech, especially in the UK. However, for more formal situations, 'Do you have..?' may be preferable. It's also worth remembering that the 'Have you got..?' form only exists in the present tense. For all other tenses, you need to use 'have' as the main verb e.g. Did you have..?/ Will you have..?