Αυτές τις μέρες είμαι πιο πιεσμένος από συνήθως, αλλά προσπαθώ να μένω ήρεμος.

Because this phrase is being used as a time expression, and in Modern Greek many expressions of time duration or time frame take the accusative.

So:

• αυτές = feminine plural accusative of αυτές / αυτοί / αυτά
• τις μέρες = feminine plural accusative of οι μέρες

Together, αυτές τις μέρες means these days.

Compare:

• αυτές οι μέρες = these days as the subject of a sentence
• αυτές τις μέρες = these days / during these days as a time expression

So in this sentence, αυτές τις μέρες is basically during these days / lately.

Πιεσμένος literally comes from the idea of being pressed or under pressure. In everyday Greek, it often means:

• stressed
• under pressure
• overloaded
• pressed for time
• mentally burdened

In this sentence, it suggests the speaker is having a more stressful or demanding period than usual.

It is very common in spoken Greek.

Yes. In Modern Greek, a very common way to form the comparative is:

• πιο + adjective

So:

• πιεσμένος = stressed / pressured
• πιο πιεσμένος = more stressed / more under pressure

This is the normal everyday pattern, just like:

• πιο κουρασμένος = more tired
• πιο ήρεμος = calmer
• πιο δύσκολος = more difficult

Greek also has some older or more formal comparative forms for certain words, but πιο + adjective is the standard pattern learners should rely on.

Because από is used in Greek to introduce the second part of a comparison, like than in English.

So:

• πιο πιεσμένος από συνήθως = more stressed than usual

Here:

• πιο = more
• από = than
• συνήθως = usually / as usual

Even though συνήθως is an adverb, the whole phrase από συνήθως is a natural way to say than usual.

Because Greek often uses είμαι + adjective to describe a person’s current state.

So:

• είμαι πιεσμένος = I am stressed / I am under pressure

This is the natural way to express the condition the speaker is in.

A verb like πιέζω means I pressure / I press, so it would not fit the same meaning unless you changed the whole structure.

For example:

• Με πιέζουν πολύ στη δουλειά. = They put a lot of pressure on me at work.
• Είμαι πολύ πιεσμένος. = I’m very stressed / under a lot of pressure.

Both are possible, but they are not exactly the same.

• να είμαι ήρεμος = to be calm
• να μένω ήρεμος = to stay/remain calm

Using μένω adds the idea of maintaining that state, especially despite pressure or difficulty.

So in this sentence, προσπαθώ να μένω ήρεμος suggests:

• I’m trying to keep calm
• I’m trying to remain calm even though things are stressful

That is why μένω works especially well here.

Να introduces a subordinate verb clause. Since Modern Greek does not use an infinitive the way English does, verbs like try, want, can, prefer, etc. are usually followed by να + verb.

So:

• προσπαθώ να μένω = I try to stay
• literally: I try that I stay

More examples:

• Θέλω να φύγω. = I want to leave.
• Μπορώ να έρθω. = I can come.
• Προσπαθώ να κοιμάμαι νωρίς. = I try to sleep early.

So after προσπαθώ, να is exactly what you expect.

In Modern Greek, after να, the verb usually appears in what learners often call the subjunctive, but for many verbs the form looks exactly like the present tense form.

So:

• μένω = I stay / I am staying
• να μένω = that I stay / to stay / to keep staying

The difference is not in a special new ending here, but in the presence of να and in the function of the clause.

Also, the present form after να gives a sense of ongoing or repeated action:

• να μένω ήρεμος = to keep staying calm / to remain calm

If the sentence used a different aspect, the nuance could change.

Because ήρεμος is an adjective, and it agrees with the subject.

Here the speaker is describing himself, so he uses:

• ήρεμος = calm (masculine singular)

If the speaker were female, it would be:

• ήρεμη

By contrast, ήρεμα is usually the adverb, meaning calmly.

Compare:

• Μένω ήρεμος. = I stay calm.
  The subject is calm.
• Μιλάω ήρεμα. = I speak calmly.
  The speaking happens calmly.

So this sentence needs the adjective, not the adverb.

Yes. Both adjectives are masculine singular, so they show that the speaker is grammatically masculine.

That means the sentence, as written, is spoken by a man or is written in a masculine form.

A feminine version would be:

Αυτές τις μέρες είμαι πιο πιεσμένη από συνήθως, αλλά προσπαθώ να μένω ήρεμη.

So adjective endings in Greek often give you information about the speaker’s gender.

Greek word order is more flexible than English word order, but the given version is very natural and neutral.

The sentence:

Αυτές τις μέρες είμαι πιο πιεσμένος από συνήθως, αλλά προσπαθώ να μένω ήρεμος.

sounds smooth and standard.

You could move things around for emphasis, for example:

• Είμαι πιο πιεσμένος από συνήθως αυτές τις μέρες...
• Αλλά προσπαθώ να μένω ήρεμος.
• Προσπαθώ όμως να μένω ήρεμος.

But the original order is probably the best version for a learner to use.

It can overlap with all of those, but in this sentence it is closest to:

• these days
• lately

It refers to the speaker’s current period of life, not necessarily a broad social trend.

So:

• Αυτές τις μέρες = these days / lately

By contrast, nowadays in English often sounds broader, like talking about the world in general. Greek can sometimes use σήμερα or other expressions in that broader sense, depending on context.

Here, αυτές τις μέρες clearly means in this period / recently.