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Outdoor is an adjective:
Fishing is an outdoor occupation.
Outdoors is a noun or an adverb:
He loved to spend time in the great outdoors.
Shall we have lunch outdoors today?
Your sentences: 1) are these sentences ok?
Have you been outdoors today? How cold is it?
How cold is it outdoors today?
How cold is outdoors today? Although this is grammatically correct, you will not often hear it said in this way. Your second sentence [with a minor change] is the most common.
outdoors 
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