Learn Hardcore Greek: Χαλαρώνω στο σαλόνι, ακόμα κι όταν είμαι πολύ αγχωμένος. - I relax in the living room, even when I am very stressed.

habitual actions: Χαλαρώνω στο σαλόνι = I (usually) relax in the living room.
actions happening now: Χαλαρώνω στο σαλόνι τώρα = I’m relaxing in the living room now.

Context decides whether you translate it with English simple present or present continuous; the Greek form itself is the same.

Do I need a reflexive idea like myself with χαλαρώνω, for example χαλαρώνω τον εαυτό μου?

No. In Greek you normally just say χαλαρώνω.

Χαλαρώνω στο σαλόνι = I relax in the living room.

The phrase χαλαρώνω τον εαυτό μου (literally I relax myself) sounds forced or unnatural in everyday speech. The simple verb χαλαρώνω already carries the meaning relax / unwind / calm down without any extra pronoun.

What exactly is στο in στο σαλόνι, and why isn’t it σε το σαλόνι?

Στο is a contraction:

σε (in, at, to) + το (the, neuter singular) → στο

Greek very often contracts σε with the definite article:

σε + τον → στον
σε + την / τη → στη(ν)
σε + το → στο
σε + τα → στα

So στο σαλόνι literally is in the living room.
Writing σε το σαλόνι is grammatically wrong in modern Greek.

Could I say σε σαλόνι without the article, or do I need στο σαλόνι?

In this context you normally need the article:

Χαλαρώνω στο σαλόνι = I relax in the living room. (a specific, known living room)

You would say σε σαλόνι only if you make it clearly indefinite with ένα:

σε ένα σαλόνι = in a living room (not a specific one)

Just σε σαλόνι without ένα or το sounds incomplete or unnatural in everyday Greek. With rooms of the house, the definite article is usually used when you mean the one in your home.

What does ακόμα κι όταν mean literally, and how does it work in the sentence?

Ακόμα κι όταν = even when.

Breaking it down:

ακόμα: usually still / yet, but in the fixed expression ακόμα και it helps mean even.
και / κι: and, but in ακόμα και it’s part of the idiom even.
όταν: when

So ακόμα κι όταν είμαι πολύ αγχωμένος = even when I am very stressed/anxious.

Grammatically, ακόμα κι όταν introduces a subordinate clause expressing a condition or situation that is surprising or contrary to expectation:
I relax … even when I’m very stressed.

Why is it κι instead of και in ακόμα κι όταν? Is there any difference in meaning?

Κι is just a phonetic variant of και. The meaning is the same.

και = and / also / even
κι is used mainly:
- before words starting with a vowel (like όταν)
- to make pronunciation smoother and faster in speech

So you can also write:

ακόμα και όταν είμαι πολύ αγχωμένος

Both ακόμα κι όταν and ακόμα και όταν are correct and mean even when.

Can I replace ακόμα with ακόμη here? Is there any difference?

Yes, you can:

Ακόμα κι όταν είμαι πολύ αγχωμένος
Ακόμη κι όταν είμαι πολύ αγχωμένος

Both are correct and mean the same thing: even when I am very stressed.

In modern Greek:

ακόμα and ακόμη are essentially synonyms.
Some people feel ακόμη is slightly more formal or written, and ακόμα a bit more colloquial, but in everyday use they are interchangeable here.

What is αγχωμένος grammatically, and how does it change for gender and number?

Αγχωμένος is a participle that functions as an adjective, meaning stressed / anxious.

It comes from the verb αγχώνομαι (to be stressed, to feel anxious).

Basic forms:

Masculine: αγχωμένος
Feminine: αγχωμένη
Neuter: αγχωμένο

Plural:

Masculine: αγχωμένοι
Feminine: αγχωμένες
Neuter: αγχωμένα

With είμαι, it works like English to be stressed:

Είμαι αγχωμένος (man)
Είμαι αγχωμένη (woman)
Είμαστε αγχωμένοι (group with at least one man)
Είμαστε αγχωμένες (all women)

In the sentence, είμαι πολύ αγχωμένος = I am very stressed / anxious (speaker is male).

What is the difference between είμαι πολύ αγχωμένος and έχω πολύ άγχος?

Both are natural, but there is a small nuance:

Είμαι πολύ αγχωμένος
- literally: I am very stressed/anxious.
- focuses on your emotional/mental state as an adjective.
Έχω πολύ άγχος
- literally: I have a lot of stress/anxiety.
- focuses on having a lot of stress as a thing (noun).

In practice, both can describe the same situation.
You might choose είμαι πολύ αγχωμένος a bit more for how you feel;
έχω πολύ άγχος a bit more for talking about the amount of stress you are under.

Why is πολύ placed before αγχωμένος? Can I say αγχωμένος πολύ instead?

Here πολύ is an adverb meaning very, so it goes before the adjective:

πολύ αγχωμένος = very stressed / very anxious

Putting it after the adjective (αγχωμένος πολύ) sounds unnatural in this context.

When πολύ is an adverb:

it does not change for gender, number, or case:
- πολύ αγχωμένος, πολύ αγχωμένη, πολύ αγχωμένο

When it is an adjective meaning much / many / a lot of, it does change:

πολύς καφές (much / a lot of coffee, masculine)
πολλή δουλειά (a lot of work, feminine)
πολύ νερό (much / a lot of water, neuter)

In your sentence it’s the invariable adverb form.

Can I change the word order and say Ακόμα κι όταν είμαι πολύ αγχωμένος, χαλαρώνω στο σαλόνι? Does it change the meaning?

Yes, that word order is completely correct:

Χαλαρώνω στο σαλόνι, ακόμα κι όταν είμαι πολύ αγχωμένος.
Ακόμα κι όταν είμαι πολύ αγχωμένος, χαλαρώνω στο σαλόνι.

Both mean the same in basic content: I relax in the living room, even when I’m very stressed.

The difference is one of emphasis:

Starting with Χαλαρώνω στο σαλόνι: you first state what you do, then add the surprising condition.
Starting with Ακόμα κι όταν είμαι πολύ αγχωμένος: you first highlight the difficult situation, then say that despite that, you relax in the living room.

Greek word order is quite flexible, and both versions are natural.