Horns of Behemoth

Yes, it becomes a minion with the Demon type, and is no longer a site.

Assuming it was in the realm prior to the turn it awakened, it is not summoning sick (since it was already under your control prior to this turn).

Yes. It is an activated ability with no cost and a threshold requirement. Like all activated abilities, you can only activate it during your Main Phase when nothing else is happening, but you are not required to activate it.

Yes. The transformed Demon provides affinity, due to the golden rule. Obviously, if the Demon dies, you will lose that affinity.

Since transform effects permanently change the characteristics of the card, as opposed to temporary modifications of characteristics (i.e. Overpower), the copy effect will take on those new characteristics. For instance, a Selfsame Simulacrum copying an Island Leviathan that has been transformed will enter play as an 8-power Monster minion that provides one water affinity.