Personally, I would have said: It is supposed to be wet and cold tomorrow OR The weather is supposed to be wet and cold tomorrow OR We're supposed to have wet and cold weather tomorrow I wouldn't actually say "It is supposed to have a wet and cold weather tomorrow". I can't think of any situations where this would be used, but maybe someone else on here could correct me on that.

Both sentences are incorrect. Some correct examples: It is supposed to be wet and cold tommorrow We are supposed to have a wet and cold day tommorrow. The weather is supposed to be cold and wet tommorrow. (all of them have exactly the same meaning)

Can't explain it gramatically but "it is" is talking about the termical sensation will carry the incoming weather. While "we're having" is talking about termical sensation you, me, all of you will feel in the current time being.