Yep, it's wrong :) "I wanna home" is ok in informal spoken English, although it is not strictly correct. And you'd never write that, except in reporting someone's speech. But "I wanna a home" is wrong in all circumstances. Are you sure that's what was said? It sounds very unnatural.

Yes. We never use "wanna" or "gonna" in any kind of written communication. They are simply how you're write out the way people in North America commonly pronounce "want to" and "going to". It's also incorrect to say "I want to (wanna) a home". The child means "I want to go home".

Oh yes indeed it is! We don't really use "wanna" as much as you think we do, and if we write the phrase in normal English, we have "I want a a home." No-one would ever say that; even in casual English it's completely unnatural.