Does it have to be 下着雨 instead of just 下雨?
I think "...你出去要带雨伞" should be accepted too but I would like to hear a native speaker's opinion. Would it sound weird to say 雨伞 instead of just 伞?
Yes, it is a bit weird in actual use. But it is definitely a valid sentence.
It does not sound weird, but it depends on area which one is more common. Both should be considered as correct.
In Sichuan / Chengdu area they say 雨伞
Personally I prefer to use 雨伞 over just 伞