You can't exert a creature unless an effect allows you to do so. Similar effects that "tap and freeze" a creature or put stun counters on a creature don't exert that creature.

You can exert Rohirrim Chargers 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.

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

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

If the exerted creature is no longer on the battlefield or the Equipment can't legally be attached to that creature, that Equipment will enter the battlefield unattached.