“Paula is moving to Chicago, where winters are cold.” is the only sentence that makes sense. The first sentence has two independent clauses not separated by anything (semicolon, period, conjunction, etc.), which is grammatically incorrect. The third sentence separates the clauses with “so”, which doesn’t make sense, either, because that indicates that the second clause was a result of the first clause, but the weather in Chicago isn’t impacted by where Paula lives. The idea is laughable! Anyways, the fourth sentence doesn’t work because it uses “however”, which is incorrect because the two clauses don’t contradict each other. Therefore, the second sentence is the only one that fits.