I want to make sure that I understand the basic usage of から.
If I say: まどをしめます。さむいですから。 Would this sound natural or stilted to native speakers?
It is perfectly grammatical, and is used regularly, but it is not in the most natural word order. から is a conjunct that heads an adverbial clause, and adverbials appear to the left of the verb phrase in Japanese. The most natural order is
The one you have splits this into two sentences, and the latter involves inversion.