You can't exert a creature unless an effect allows you to do so. Similar effects that "tap and freeze" a creature (such as that of Decision Paralysis) don't exert that creature.

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

Because damage remains marked on a creature until it's removed as the turn ends, the damage Pride Sovereign takes during combat may become lethal if other Cats you control leave the battlefield later in the turn.

Some cards in the Hour of Devastation set let you exert a creature as a cost to activate one of its abilities. You can exert it to pay that cost even if you've already exerted it earlier in the turn. Exerting it multiple times will keep it tapped only during your next untap step.