Learn Hardcore Polish: Migający przycisk w windzie uspokaja zdenerwowane dzieci. - The blinking button in the elevator calms the nervous children.

Forms like w winda or w windą are grammatically incorrect in this context.
Windą is instrumental (e.g. jadę windą – I’m going by elevator), not used for simple location.

Why is the verb uspokaja (3rd person singular) and not uspokajają (3rd person plural)?

The verb must agree with the subject, which is migający przycisk.

Subject: migający przycisk – singular, masculine
Verb: uspokaja – 3rd person singular

So:

Migający przycisk … uspokaja zdenerwowane dzieci.
The flashing button calms the nervous children.

If the subject were plural, the verb would be plural too, e.g.:

Migające przyciski uspokajają zdenerwowane dzieci.
(Flashing buttons calm the nervous children.)

Here, because there is only one przycisk, uspokaja is correct.

What tense/aspect is uspokaja, and how is it different from uspokoi?

Uspokaja is:

present tense
3rd person singular
imperfective aspect
from the verb uspokajać (to calm, to be calming)

It can mean:

a general truth/habit:
Migający przycisk uspokaja dzieci. – A flashing button calms children (in general).
something happening now:
Right now, the flashing button is calming the kids.

Uspokoi is perfective, from uspokoić, and usually refers to a single, completed action, typically in the future:

Migający przycisk uspokoi dzieci. – The flashing button will calm the children (once, at some point).

So:

uspokaja – ongoing, repeated, or habitual action
uspokoi – one complete calming event, often understood as future

Why is it zdenerwowane dzieci and not zdenerwowani dzieci?

The noun dzieci (children) is grammatically non‑masculine plural, and adjectives agreeing with it take the neuter plural form, not the masculine personal form.

So you say:

małe dzieci (small children)
zmęczone dzieci (tired children)
zdenerwowane dzieci (nervous/upset children)

Forms like zdenerwowani dzieci are wrong, because zdenerwowani is masculine personal plural (used with men or mixed groups where at least one male is included, e.g. zdenerwowani chłopcy – upset boys).

Even if in reality the group of children is mixed or mostly boys, grammatically dzieci still takes neuter plural agreement: zdenerwowane dzieci.

What case is zdenerwowane dzieci here, and how do we know?

In this sentence, zdenerwowane dzieci is in the accusative plural, because it is the direct object of the verb uspokaja (it answers “whom?” / “what?”):

Co / kogo uspokaja? – What/whom does it calm?
uspokaja zdenerwowane dzieci – calms the nervous children

For dzieci, the nominative plural and accusative plural have the same form:

Nominative plural: dzieci
Accusative plural: dzieci

The function in the sentence (object vs subject) tells you the case, not the ending.

Can the word order be different, like W windzie migający przycisk uspokaja zdenerwowane dzieci?

Yes. Polish word order is relatively flexible. All of these are grammatically correct:

Migający przycisk w windzie uspokaja zdenerwowane dzieci.
W windzie migający przycisk uspokaja zdenerwowane dzieci.
Migający przycisk uspokaja w windzie zdenerwowane dzieci. (less natural, but possible)

The differences are mostly in emphasis and information structure:

Starting with Migający przycisk makes the button the main topic.
Starting with W windzie highlights the location first.
Putting w windzie near przycisk makes it more clearly modify the button (button in the elevator), not the children.

The given sentence is a very natural, neutral word order: [Subject + location phrase] + [verb] + [object].

Could we say migający guzik w windzie instead of migający przycisk w windzie? What’s the difference?

You can say migający guzik w windzie, and people will understand you, but there is a nuance:

przycisk – a functional button, like the ones you press on devices, control panels, lifts.
guzik – primarily a clothing button (on shirts, coats), but also used colloquially for many small “buttons” you press.

In everyday speech, many people casually say guzik for a button on a device, so guzik w windzie isn’t wrong.
However, przycisk is more precise and standard for a button on a machine or panel, so migający przycisk w windzie sounds a bit more “correct” or neutral in standard Polish.

There is no a or the in Polish. How do we know if it’s “a flashing button” or “the flashing button”?

Polish simply has no articles, so migający przycisk can correspond to:

a flashing button
the flashing button

Which one is intended depends on context, not grammar. You can tilt it toward “the” or “this” with words like:

ten migający przycisk – this / that flashing button
jakiś migający przycisk – some flashing button

In isolation, Migający przycisk w windzie uspokaja zdenerwowane dzieci is most naturally translated into English with the:
The flashing button in the elevator calms the nervous children, but grammatically the sentence itself doesn’t force a vs the.

Is there any difference between zdenerwowane dzieci and something like nerwowe or zestresowane dzieci?

Yes, there are slight nuances:

zdenerwowane dzieci – children who are upset, agitated, worked up, often due to some recent event (angry, scared, crying, etc.).
nerwowe dzieci – nervous/excitable children in general; it can suggest a character trait (they get nervous easily), not necessarily just a temporary state.
zestresowane dzieci – stressed children, often due to pressure, expectations, or ongoing stress (exams, problems at home, etc.).

In your sentence, zdenerwowane dzieci focuses on the idea that the children are currently upset or on edge, and the flashing button helps to calm them down.