Learn Hardcore Greek: Η μουσική με βοηθάει να ησυχάζω όταν έχω άγχος. - Music helps me to calm down when I am stressed.

In Modern Greek, these weak object pronouns usually go right before the verb (or attached to the end in some cases), not after it as in English. So you say:

Η μουσική με βοηθάει (correct)
not
Η μουσική βοηθάει με (wrong – that sounds like with music).

Also, με here belongs to βοηθάει (the music helps me), not to ησυχάζω. It means music helps me (so that I can) calm down, not music helps to calm me.

Why do we have βοηθάει here? Is there a difference between βοηθάει, βοηθά, βοηθάω, and βοηθώ?

All of these are present‑tense forms of the verb βοηθάω / βοηθώ (to help). In practice:

βοηθάει and βοηθά are both 3rd person singular: he/she/it helps.
βοηθάω and βοηθώ are 1st person singular: I help.

The longer forms (βοηθάω, βοηθάει) are slightly more colloquial; the shorter ones (βοηθώ, βοηθά) can feel a bit more compact or formal, but all are widely used and correct.

So you could also hear:

Η μουσική με βοηθά να ησυχάζω
with no real change in meaning.

Why is it να ησυχάζω and not να ησυχάσω? Aren’t both to calm down?

Greek makes a clear distinction between imperfective and aorist aspect in the να‑forms:

να ησυχάζω (imperfective): calming down as an ongoing, repeated, or habitual process.
να ησυχάσω (aorist): calming down as a single, completed event.

In Η μουσική με βοηθάει να ησυχάζω όταν έχω άγχος, we are talking about a general habit: whenever I feel anxious, music helps me calm down (in general, repeatedly). For this habitual meaning, να ησυχάζω is the natural choice.

If you said Η μουσική με βοηθάει να ησυχάσω, it would sound more like music helps me (this time) to finally calm down, focusing on one result rather than a general pattern.

What is the nuance of ησυχάζω here? How is it different from ηρεμώ or χαλαρώνω?

All three can be translated as to calm down / relax, but they have slightly different flavors:

ησυχάζω:
Literally to become quiet, to stop noise or agitation. It can be physical (a place becomes quiet) or emotional (a person settles down). In this sentence it suggests settling down, becoming less agitated.
ηρεμώ:
More about inner calm, from ήρεμος (calm). It focuses on emotional tranquility: to calm down emotionally.
χαλαρώνω:
Closer to to relax, to unwind, often used for taking it easy, resting, not being tense.

Your sentence would also sound very natural as:

Η μουσική με βοηθάει να ηρεμώ όταν έχω άγχος.
Η μουσική με βοηθάει να χαλαρώνω όταν έχω άγχος.

The original with ησυχάζω leans slightly more toward settling down / quieting down.

Why do we say Η μουσική with the article Η? Could we just say Μουσική με βοηθάει να ησυχάζω…?

In Greek, the definite article (ο, η, το) is used more often than the in English, especially with:

Abstract nouns (η μουσική, η αγάπη, η τέχνη)
Things spoken of in a general sense (η μουσική = music in general)

So Η μουσική με βοηθάει… is the most natural way to say Music helps me… in general.

You could drop the article (Μουσική με βοηθάει να ησυχάζω), but that sounds marked or stylistic (for example, in headlines, slogans, or poetic style). In everyday speech, Η μουσική… is strongly preferred.

Why is it όταν έχω άγχος and not something like όταν είμαι αγχωμένος or όταν αγχώνομαι?

All three are possible, but they have slightly different nuances:

όταν έχω άγχος:
Literally when I have anxiety / when I have stress. Very common and neutral, can be used for both temporary and more general anxiety.
όταν είμαι αγχωμένος:
When I am anxious / stressed (out). This describes your state as an adjective.
όταν αγχώνομαι:
When I get anxious / when I become stressed. This is a verb and emphasizes the process of becoming anxious.

Your sentence uses έχω άγχος because it is a standard, everyday phrase for being anxious and sounds perfectly natural in a general, habitual statement.

Can I change the word order? For example, can I say Με βοηθάει η μουσική να ησυχάζω or Η μουσική βοηθάει με?

Word order is more flexible in Greek than in English, but the position of weak pronouns is quite fixed.

These are all correct, just with different emphasis:

Η μουσική με βοηθάει να ησυχάζω… (neutral order)
Με βοηθάει η μουσική να ησυχάζω… (slight emphasis on me)
Η μουσική με βοηθά να ησυχάζω… (shorter form of the verb)

However, Η μουσική βοηθάει με is wrong in this meaning:

βοηθάει με would be understood as helps with, where με is the preposition with, not the pronoun me.
The unstressed pronoun με must attach to the verb as με βοηθάει, not follow it as a separate με.

What exactly is να doing in με βοηθάει να ησυχάζω? Is it like English to?

να is a particle that introduces the subjunctive in Modern Greek. The structure here is:

με βοηθάει = it helps me
να ησυχάζω = (for) me to calm down

So με βοηθάει να ησυχάζω corresponds to it helps me (to) calm down.

After verbs like βοηθάω, θέλω, προσπαθώ, Greek normally uses να + verb, not an infinitive (Modern Greek no longer has an infinitive form). You cannot drop the να here; με βοηθάει ησυχάζω would be ungrammatical.

What kind of noun is άγχος? Is it countable, and what are its main forms?

άγχος is a neuter noun. Basic forms:

το άγχος – the anxiety / stress (nominative–accusative singular)
του άγχους – of the anxiety / stress (genitive singular)

In everyday language it usually behaves like an uncountable noun, similar to English stress:

έχω άγχος – I am stressed / I feel anxious.

There is a plural τα άγχη, but it is rare and sounds technical or literary (for example, in psychology or formal writing). In normal speech you mostly see the singular άγχος.

How do you pronounce άγχος, especially the γχ?

άγχος is pronounced approximately:

[Á-nghos], with the stress on the first syllable.

Details:

ά is like a in father.
γ before χ is pronounced as a ng sound [ŋ].
χ is a voiceless kh sound, similar to the ch in German Bach or Scottish loch.
So together γχ is [ŋx], a combination of ng
- kh: ά-γχος → /ˈaŋ.xos/.

For many English speakers, aiming for something close to ANG‑khos (with a rough kh) is a good approximation.