They are both correct. If you say 'I'd like to invite you for dinner.' I would understand this to mean this as an invitation to my home. It is a reduced form of 'I'd like to invite you [to my house] for dinner'. This is quite acceptable, and quite understandable. If you say 'I'd like to invite you to dinner', this could mean either at home or at a restaurant.

I think the correct one is I invite you FOR dinner

Both of these are perfectly correct in American English. The choice between one or the other is simply dialect, and I don't even think there are strong regional preference for one or the other any more.

The correct preposition is "to". Absolutely.