Let's not forget that the language is still called English (not "American") and the original country where it all started out still loves to use "I have got" to express possession - as in "This room has got a lovely view". In a typical North American, pragmatic way, it has been shortened to "have/has" with the same meaning - as in "They have a new car"; but it doesn't give us the right to claim the original is wrong. The British version goes as far as "have to" being "have got to". And this particular form is not exactly new to North America: you're much more likely to say "Gotta go/run to the bank"("I have got to go/run...") than "I have to go/run to the bank".
In Canada, I've noticed that both British and American English are acceptable and used randomly. Add the French influence - and it becomes a unique, colorful, local version. Officially, it's 28 degrees outside (°C); but the pool is 80 (°F), so it's warm enough to swim. This duality is probably not even noticed by anyone other than linguists and newcomers who try to say they are 1.70m tall - and after a few blank stares switch to 5'7". And then the rain falls in millimeters, and the snow accumulates in centimeters. How much fun is that?
Back to "have" and "have got":
- "America's Got Talent"
- "You've Got A Friend" (James Taylor)
- You've got to (gotta) be kidding me!
- "What have I got to do / To make you love me?" (Elton John, "Sorry Seems To Be The Hardest Word")
GET - GOT - GOT(TEN)
This one is a separate verb, which can mean pretty much anything. I often tell my students - only half-kidding - to use "get" when they can't come up with the verb they need. My personal suspicion is that "get-got-got" was transformed into "get-got-gotten" in America, just due to its similarity to "forget-forgot-forgotten"; but it's a hunch, not a rule.
In everyday communication, most commonly, "get" is used to express:
- "obtain": one goes to the kitchen to get some water, or needs to get some sleep, or to get enough milk to provide calcium to a growing body ("Got milk?");
- "understand": some get it - and some don't, or you heard a joke - but didn't get it;
- "become": you get really fluent if you practice a new language, you get a promotion or a raise at work, you get better after an illness;
All right, I hope you got all this straight, because I gotta go now...