You can exert Hydra Trainer 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 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 always activate an ability that will cause a creature to adapt. As that ability resolves, if the creature has a +1/+1 counter on it for any reason, you simply won't put any +1/+1 counters on it.

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.

You don't choose a target for Hydra Trainer's first ability at the time it triggers. Rather, a second "reflexive" ability triggers when you exert Hydra Trainer this way. You choose a target for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.

The value of X is determined only once, as the reflexive triggered ability resolves.

If a creature somehow loses all of its +1/+1 counters, it can adapt again and get more +1/+1 counters.