Όταν έχω άγχος, προσπαθώ να βοηθήσω τον εαυτό μου να μείνει ήρεμος.

A helpful way to see it is in layers:

• Όταν έχω άγχος = a time clause, when I have stress / when I feel anxious
• προσπαθώ = I try
• να βοηθήσω = to help
• τον εαυτό μου = myself
• να μείνει ήρεμος = to stay/remain calm

So the structure is basically:

When I feel anxious, I try [to help myself [to remain calm]].

Greek often builds sentences like this with one να clause inside another.

Because έχω άγχος is a very normal Greek expression. Literally it is I have anxiety/stress, but idiomatically it often means I am anxious or I feel stressed.

This is very common in Greek:

• έχω άγχος = I am stressed / anxious
• έχω φόβο or more naturally φοβάμαι = I am afraid
• έχω υπομονή = I have patience

So even if English prefers an adjective, Greek may prefer a noun with έχω.

Because Greek often leaves out the article with abstract nouns when speaking in a general way.

So:

• έχω άγχος = I am anxious / I have stress
• not usually έχω το άγχος in this meaning

If you said το άγχος, it would sound more like the anxiety, referring to a specific anxiety already known in the conversation.

Because Greek usually does not need subject pronouns. The verb ending already tells you who the subject is.

• έχω = I have
• προσπαθώ = I try

So εγώ is usually omitted unless you want emphasis or contrast:

• Εγώ προσπαθώ, αλλά αυτός όχι.
  I try, but he doesn’t.

In your sentence, no emphasis is needed, so leaving εγώ out is completely normal.

Όταν means when. In this sentence it introduces a general or repeated situation:

• Όταν έχω άγχος... = When(ever) I feel anxious...

Because this is a habitual idea, Greek uses the present tense:

• όταν έχω άγχος = when I have anxiety / whenever I feel anxious

If you were talking about a single future event, Greek could use a different structure, but here the meaning is general: this is what I do whenever that situation happens.

Modern Greek does not usually use an infinitive like English to help or to stay. Instead, it uses να plus a verb form.

So:

• να βοηθήσω = roughly to help
• να μείνει = roughly to stay / remain

There are two να words because there are two linked actions:

• προσπαθώ να βοηθήσω... = I try to help...
• ...τον εαυτό μου να μείνει ήρεμος = ...myself to remain calm

So one να clause is nested inside another.

After να, Greek chooses between different aspect forms. Here βοηθήσω is the aorist/perfective form.

That means the helping is viewed as a single whole action or goal:

• προσπαθώ να βοηθήσω τον εαυτό μου = I try to help myself

If you used the imperfective form, it would sound more like an ongoing or repeated process. In this sentence, Greek naturally prefers the perfective idea: trying to achieve the result of helping oneself.

So βοηθήσω here does not mean past tense. After να, it is not a past form; it is a subjunctive-type perfective form.

να μείνει is again the perfective form, from μένω.

• μένω = I stay / remain
• να μείνει = to stay / remain, seen as a complete result

This fits the idea of trying to get oneself into a calm state and keep that calm state.

Compare:

• να είναι ήρεμος = to be calm
• να μένει ήρεμος = to keep staying calm, more ongoing/habitual
• να μείνει ήρεμος = to remain/stay calm, focusing on the result or whole event

So the original sentence is very natural.

Because Greek usually forms reflexive expressions with ο εαυτός plus a possessive pronoun:

• τον εαυτό μου = myself
• τον εαυτό σου = yourself
• τον εαυτό του/της = himself/herself

Literally, it is something like my self.

Greek normally does not use a simple object pronoun alone for this meaning. So με βοηθάω would not be the normal way to say I help myself. The standard Greek reflexive expression is βοηθώ τον εαυτό μου.

Because in Greek, εαυτός behaves like a normal noun, and nouns normally take an article in this kind of structure.

So:

• τον εαυτό μου = myself

Also, here it is in the accusative because it is the direct object of βοηθήσω:

• βοηθήσω ποιον; = help whom?
• τον εαυτό μου = myself

So the article is not optional here; it is the normal Greek way to build the reflexive phrase.

Because it agrees with the understood subject of να μείνει, which is ο εαυτός μου.

Even though you see τον εαυτό μου earlier in the sentence as an object of βοηθήσω, inside the smaller clause the understood idea is:

• ο εαυτός μου να μείνει ήρεμος
  myself to remain calm

Since εαυτός is masculine singular, the adjective is also masculine singular:

• ήρεμος

This is why it is not:

• ήρεμη, which would be feminine
• ήρεμο, which would be neuter

It can lean a little toward both, depending on context.

With μένω, Greek often expresses the idea of:

• remaining in a state
• staying in a state
• ending up in that state and continuing there

So here να μείνει ήρεμος is very naturally understood as:

• to stay calm
• to remain calm

In context, it can also carry the idea of getting oneself calm and keeping that calmness, which is why it works well after βοηθήσω τον εαυτό μου.

Yes. Greek word order is fairly flexible, though some orders are more natural than others.

Your sentence:

• Όταν έχω άγχος, προσπαθώ να βοηθήσω τον εαυτό μου να μείνει ήρεμος.

You could also say:

• Προσπαθώ να βοηθήσω τον εαυτό μου να μείνει ήρεμος όταν έχω άγχος.

Both are understandable. The original version sounds very natural because it sets up the situation first:

• When I feel anxious, ...

That is a common and clear way to organize the sentence.

Yes. A few natural alternatives are:

• Όταν αγχώνομαι, προσπαθώ να βοηθήσω τον εαυτό μου να μείνει ήρεμος.
  Using αγχώνομαι = I get anxious / I feel anxious

• Όταν έχω άγχος, προσπαθώ να ηρεμήσω.
  When I feel anxious, I try to calm down.
  This is shorter and less explicitly reflexive.

• Όταν έχω άγχος, προσπαθώ να βοηθήσω τον εαυτό μου να ηρεμήσει.
  ...to help myself calm down.

Your original sentence is perfectly natural, but these alternatives show slightly different shades of meaning:

• να μείνει ήρεμος = remain/stay calm
• να ηρεμήσει = calm down