Yes, they are both correct. I have = I've got

Yes. I believe "I've got.." is more American sounding, but I'm American and I'm just as likely to use either one.

"I have a mother" rhymes better and sounds fine to my ears. "I've got a mother" sounds vulgar. Discrimination between elegant and vulgar language and attitude is, unfortunately, not a skill everybody possesses. It's worth to train one's self in order to distinguish vulgar attitude and language from noble language, it improves the whole society and makes communication joyful and pleasant.