Bingley, Hyla and Marianne have all pretty much gotten the idea of how the two are used. With not infrequent exceptions, we in the US do not generally use "got" to mean "have". Notwithstanding some pretty obvious usages, like, "I've got music, I've got rhythm", "I've got a loverly bunch of coconuts", "He's got the whole world in His hands", which is the possessive use, or "I've got you under my skin" which is not. On the other hand, if/when we do use "got" for the possessive, it tends to be as an emphatic expression. One would generally say, "I have two apples" rather than "I've got two apples"; but I think we would tend more to "He's got a gun pointed at you!" but "He has a gun stored in the garage." Then there's the fact that the exceptions that come to my mind are from songs; I would agree that "gotten" is unmusical and "have" is a problem for musical purposes.

So to sum up, Americans tend on the whole not to use "got" for "have" as the normal verb of possession. They use "got" as the past tense, or preterite, of "get", but generally use "gotten" as the past participle to form the perfect tense, rather than "got", which is British usage. To the preference for "have" over "got", there are exceptions, notably for emphasis, for euphony. I'm sure there will be more that will occur to you out there.