If an exerted permanent is already untapped during your next untap step (most likely because an effect untapped it), exert's effect preventing it from untapping expires without having done anything.

If you gain control of another player's permanent until end of turn and exert it, and then that player regains control of it, it will untap during that player's untap step.

You can exert Anep as you declare it as an attacking creature. You can't do so later in combat, and creatures put onto the battlefield attacking can't be exerted. Any abilities that trigger on exerting an attacking creature will resolve before blockers are declared.

You pay all costs and follow all timing rules for cards played with the permission granted by the ability that triggers when you exert Anep. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.