RAW, I believe that the Unseen Servant cannot take Actions, but can drop plates.

This spell creates an invisible, mindless, shapeless, Medium force that performs simple tasks at your command until the spell ends.

The spell does not say that it's a creature, or other being capable of taking actions. Things do what they say they do; note that the Find Familiar spell says that the summoned thing can take actions.

Your familiar acts independently of you, but it always obeys your commands. In combat, it rolls its own initiative and acts on its own turn. A familiar can't attack, but it can take other actions as normal.

However, the servant is capable of doing simple tasks that a human servant can do:

Once on each of your turns as a bonus action, you can mentally command the servant to move up to 15 feet and interact with an object. The servant can perform simple tasks that a human servant could do, such as fetching things, cleaning, mending, folding clothes, lighting fires, serving food, and pouring wine. Once you give the command, the servant performs the task to the best of its ability until it completes the task, then waits for your next command.

I would argue that dropping dishes is something that servants do, and is essentially instantaneous (no hesitation, because the servant is mindless). This should be able to happen during the bonus action on the player's turn, unless there's a reason otherwise (i.e. the activity takes too long).

Result

Causing a distraction by dropping plates is certainly something that the servant should be able to do on the player's bonus action. However, as the DM you get to decide what happens after that. When I DM, I personally use the Rule of Cool - in this case, I'd probably grant a wisdom save or intelligence check for the creature (at an appropriate DC) on the first set of dishes, and then either ignore future sets, or with advantage. If the player starts running around with stacks of dishes otherwise, remind them that dishes cost gold, and people don't like their dishes being stolen.

No

Being hit, and taking damage, are not simultaneous events, though they are very closely linked in time. We can see this, from the wording of the "Resolve the attack step":

Resolve the attack. You make the attack roll. On a hit, you roll damage, unless the particular attack has rules that specify otherwise. Some attacks cause special effects in addition to or instead of damage.

Taking damage, is a contingent, but separate, event to being hit. This is made clearer by examining the rules for reactions:

A reaction is an instant response to a trigger of some kind, which can occur on your turn or someone else's.

As being hit, and taking damage are separate triggerable events (the shield spell trigger happens after being hit, but before damage happens as an example), they cannot be simultaneous.

We can see this by examining the Xanathar's Guide to Everything rules for simultaneous effects:

If two or more things happen at the same time on a character or monster's turn, the person at the game table-whether player or DM-who controls that creature decides the order in which those things happen. For example, if two effects occur at the end of a player character's turn, the player decides which of the two effects happens first.

If this rule was applied, and we took being hit, taking damage and the reaction as all being simultaneous, then the creature whose turn it is, would get to decide the timing and resolution of all the effects. In particular, they would get to choose, that the damage happened from the attack, before the Shield spell resolved, allowing for the triggering of effects that occur on damage, while enabling the Shield spell to then cancel the hit, and prevent triggers that rely on hitting.

Clearly, this is an absurd situation, if we treat damage and being hit as being simultaneous events.