Το μαγείρεμα με βοηθάει να χαλαρώνω το βράδυ.

It is a noun, not a finite verb.

Το μαγείρεμα comes from the verb μαγειρεύω (to cook) and means cooking as an activity. In English, cooking can look like a verb form, but here the Greek word is clearly being used as a noun:

• το μαγείρεμα = cooking
• Το μαγείρεμα με βοηθάει... = Cooking helps me...

So grammatically, το μαγείρεμα is the subject of the sentence.

Greek uses the definite article much more often than English.

With general activities, abstract ideas, and nouns used in a broad sense, Greek often includes the article:

• το μαγείρεμα = cooking
• η μουσική = music
• το διάβασμα = studying / reading

So το μαγείρεμα does not necessarily mean the cooking in the English sense. In this sentence, it simply means cooking as a general activity.

Because βοηθάω / βοηθώ usually takes a direct object in Modern Greek, so you use the accusative pronoun:

• με = me
• σε = you
• τον / την = him / her
• μας = us

So:

• με βοηθάει = it helps me

For an English speaker, this can feel surprising because English uses help + object too, but Greek learners sometimes expect μου because it can look like an indirect-object idea. In this case, though, με is the normal choice.

Yes. βοηθάει and βοηθά are both common forms meaning he/she/it helps.

In your sentence:

• Το μαγείρεμα με βοηθάει = Cooking helps me

You may also hear or see:

• Το μαγείρεμα με βοηθά

Same meaning. The longer form -άει is very common in everyday Greek, and the shorter form is also perfectly normal.

Because Greek does not use an infinitive the way English does.

English says:

• helps me relax

Greek uses να + verb instead:

• με βοηθάει να χαλαρώνω

So here να introduces the action that cooking helps with.

A very useful way to think of it is:

• English: to relax / relax
• Greek: να χαλαρώνω

This is a question of aspect, which is very important in Greek.

• να χαλαρώνω = imperfective
• να χαλαρώσω = perfective

In this sentence, να χαλαρώνω is used because the meaning is habitual / repeated / ongoing:

• cooking helps me relax in the evening
• it is something that generally happens
• it is part of a routine

If you said να χαλαρώσω, it would sound more like to relax once, to manage to relax, or to reach a relaxed state in a more single-event sense.

So να χαλαρώνω fits better with the general idea of an evening habit.

It looks like the present tense form, but after να it is functioning as a subjunctive form.

So in this sentence:

• χαλαρώνω is not simply I relax
• να χαλαρώνω means something like to relax / that I relax, with imperfective aspect

This is very common in Greek: the form after να often looks the same as a present-tense form, but its job in the sentence is different.

Here το βράδυ means in the evening or at night, depending on context.

So:

• Το μαγείρεμα με βοηθάει να χαλαρώνω το βράδυ
  = Cooking helps me relax in the evening / at night

Greek very often uses the accusative like this for time expressions without needing a preposition.

Common examples:

• το πρωί = in the morning
• το μεσημέρι = at noon / in the midday period
• το απόγευμα = in the afternoon
• το βράδυ = in the evening / at night

Because many Greek time expressions are normally used with the article.

So το βράδυ is the standard expression, just like:

• το πρωί
• το απόγευμα
• το βράδυ

You should learn these as set expressions. Even though English often says just at night or in the evening, Greek usually keeps the article.

Yes, you could, and it is grammatical.

But there is a slight difference in feel:

• Το μαγείρεμα = cooking as an activity, more compact and noun-like
• Το να μαγειρεύω = the act of cooking / cooking myself, more explicitly verbal

So both are possible, but Το μαγείρεμα με βοηθάει... sounds very natural and concise.

Not completely. Greek word order is fairly flexible, although some orders sound more natural than others.

Your sentence is very natural:

• Το μαγείρεμα με βοηθάει να χαλαρώνω το βράδυ.

You could also move το βράδυ for emphasis:

• Το βράδυ, το μαγείρεμα με βοηθάει να χαλαρώνω.

But the clitic pronoun με normally stays before the verb:

• με βοηθάει not
• βοηθάει με

So the sentence has some flexibility, but not total freedom.

Not always. Its basic idea is loosen / relax, and the exact meaning depends on context.

Here it clearly means:

• I relax / unwind

But in other contexts it can also mean things like:

• to loosen
• to become less strict / less tense

In your sentence, the meaning is definitely the personal one: to relax, to unwind.