Hello Alex, hi Minka,
I don't know about America but in England we do actually use both and I think it's a matter of what you're actually talking about (the main point being
yourself or what
you have) "I have got a dog" I would use when I'm just talking about myself. "I have a dog" I would often use if I want to make a point out of what I have.
Let me give you some examples.
"You can't stand guard because you haven't got a weapon" "Yes I can, I
have a dog"
"Why can't you just go on holiday and enjoy yourself?" "Because I
have a dog", now using "I have" you would emphasis the word "have"
"I'm such a lucky person,
I've got a dog,
I've got money and
I've also got a swimming pool" now here you would slightly emphasis the word "I've"
It's definetly one of those things you have to get a feel for

.
Wish you well,
Caro