Yes, you would make the check during your turn, and if successful, you have communicated to the animal that it is to attack that target. The rest works just like if you talked to another humanoid and told them to attack that person. If they choose to comply (which the trick ensures the animal will), then they must wait until their own turn to act on the command.

As for how long they keep attacking on one command, the RAW is somewhat more ambiguous on that point, but the description of the "Down" trick seems to imply that the "Attack" trick lasts until the enemy being attacked is taken down or until something such as the "Down" trick or severe injury motivates the animal to fall back.

Down (DC 15) The animal breaks off from combat or otherwise backs down. An animal that doesn’t know this trick continues to fight until it must flee (due to injury, a fear effect, or the like) or its opponent is defeated.

In general, yes. But there are exceptions

The general rule is outlined in the PHB:

Sometimes an attacker wants to incapacitate a foe, rather than deal a killing blow. When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out. The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable.

It lists no conditions for this to be true, so, in the absence of something specific saying that you cannot, you can knock out the creature.

However, per this Q&A there are at least a couple creatures that do not appear to be able to be knocked unconscious because of effects that take place when they get reduced to 0 HP.

Creatures with immunity to being unconscious

This one is pretty obvious. If a creature is immune to being unconscious then it can't become unconscious. Earth, Air, and other elementals often have this.

See this Q&A thread for more exploration of this topic.

Most Creatures with the Constructed Nature feature

Monster Manual pg.19 covers this.

Constructed Nature.

An animated object doesn't require air, food, drink, or sleep. The magic that animates an object is dispelled when the construct drops to 0 hit points. An animated object reduced to 0 hit points becomes inanimate and is too damaged to be of much use or value to anyone.

In this case, when a creature with this feature gets knocked to 0 HP, they simply lose the magic and they become inanimate.

Golems

Blind Obedience.

When its creator or possessor is on hand to command it, a golem performs flawlessly. If the golem is left without instructions or is incapacitated, it continues to follow its last orders to the best of its ability. When it can't fulfill its orders, a golem might react violently-or stand and do nothing. A golem that has been given conflicting orders sometimes alternates between them.

A golem can't think or act for itself. Though it understands its commands perfectly, it has no grasp of language beyond that understanding, and can't be reasoned with or tricked with words. (Monster manual p. 167)