Does the expression ' a bad spell of wether' refer to the fact that someone has made a spelling mistake?
Without context, that is not possible to say.
I'm no teacher
I think it's simply a misapplication of "a spell of bad weather".
Yes, and in this case, "spell" means "period of time" or perhaps "report."
That of course is the obvious answer. But the OP may have heard it in a particular context which might alter that assumption. As a phrase it is certainly not one in general use referring to a person having problems with spelling.