All of them can mean the same thing, yes. I have a pig is grammatically correct and is said everywhere. I've got a pig is also grammatically correct, but said mostly in Britain and not in America. I got a pig can mean the same thing as I have a pig in non-grammatical American slang, but is never said in Britain. Here in britain, it is grammatical, however to say I got a pig, meaning I bought/recieved a pig.
Hope that makes sense :)