Learn Swedish: Han gömmer pistolen bakom sig. - He hides the gun behind him.

Because the person he is hiding the gun behind is himself. In Swedish, when a pronoun inside a prepositional phrase (like bakom) refers back to the subject of the same clause (here, han), you must use the reflexive pronoun sig. If you wrote bakom honom, it would mean he is hiding the gun behind a completely different man.

Does the sig make gömma (to hide) a reflexive verb in this sentence?

No. Here, the object being hidden is pistolen (the gun). The sig belongs entirely to the prepositional phrase bakom sig (behind himself) to show the location. Gömma only acts as a reflexive verb when a person is hiding themselves.

If he were hiding himself behind the gun, how would the sentence change?

You would say Han gömmer sig bakom pistolen. In that case, sig acts as the direct object of the verb (he hides himself), and the prepositional phrase is bakom pistolen. Comparing these two sentences is a great way to see the difference between a reflexive verb and a spatial reflexive pronoun!

Could I change the word order and say Han gömmer bakom sig pistolen?

No, that sounds very unnatural. In a basic Swedish main clause, the direct object (pistolen) generally needs to come before the prepositional phrase (bakom sig) that describes the location.