Θέλω να γίνω καλά σύντομα, γιατί δεν αντέχω άλλο αυτή τη ζάλη και αυτόν τον βήχα.

Because γίνομαι καλά means to get well / to become well, while είμαι καλά means to be well.

So:

• Θέλω να γίνω καλά σύντομα = I want to get well soon
• Θέλω να είμαι καλά = I want to be well

In this sentence, the speaker is sick right now, so the idea is a change of state: from unwell to well. That is why γίνω is the natural choice.

Here καλά functions adverbially, meaning something like well.

Compare:

• Είμαι καλά. = I am well.
• Θέλω να γίνω καλά. = I want to get well.

Even though καλά is also the neuter plural or adverbial form of καλός, in expressions about health it is the normal form.

So you should learn γίνομαι καλά and είμαι καλά as very common set phrases.

In Greek, verbs like θέλω are often followed by να plus a verb form to express want to..., in order to..., or other subordinated actions.

So:

• Θέλω να γίνω καλά = I want to get well

This is one of the most basic Greek patterns:

• Θέλω να φύγω. = I want to leave.
• Θέλει να φάει. = He/She wants to eat.
• Θέλουμε να πάμε. = We want to go.

English uses to; Greek usually uses να.

Γίνω is the aorist subjunctive form of γίνομαι.

After να, Greek often uses either:

• the present subjunctive for ongoing/repeated action
• the aorist subjunctive for a single whole action/event

Here, να γίνω καλά refers to becoming well as a complete event, so the aorist form is natural.

Compare:

• Θέλω να γίνω καλά. = I want to get well.
• Θέλω να γίνομαι καλύτερα κάθε μέρα. = I want to be getting better every day.
  This would be more ongoing/habitual and is much less likely in this exact context.

Σύντομα means soon.

In this sentence:

• Θέλω να γίνω καλά σύντομα = I want to get well soon

Greek word order is more flexible than English, so σύντομα could appear in different places, for example:

• Θέλω να γίνω σύντομα καλά
• Σύντομα θέλω να γίνω καλά

But the original order sounds very natural.

Here γιατί means because.

So:

• ..., γιατί δεν αντέχω άλλο ... = ..., because I can't stand ... any longer

Important: γιατί can also mean why in questions:

• Γιατί έφυγες; = Why did you leave?

So the same word can mean:

• because
• why

You understand it from context and punctuation.

Δεν αντέχω άλλο means I can’t stand it any longer / I can’t take it anymore.

Breakdown:

• δεν = not
• αντέχω = I endure, tolerate, stand
• άλλο = more / any longer / anymore

Together, δεν ... άλλο often gives the idea of no longer, not anymore, or any longer.

Very common examples:

• Δεν μπορώ άλλο. = I can’t anymore.
• Δεν αντέχω άλλο. = I can’t take it anymore.

Because here άλλο does not directly describe ζάλη or βήχα. It works adverbially with the verb αντέχω.

So:

• δεν αντέχω άλλο = I can’t stand it any longer

It means any longer / anymore, not another dizziness or another cough.

If άλλος were agreeing with a noun, then it would change by gender/case:

• άλλη ζάλη = another dizziness
• άλλον βήχα = another cough

But that is not what is happening here.

This is normal Greek. A demonstrative like αυτός, αυτή, αυτό is commonly used together with the definite article.

So:

• αυτή τη ζάλη = this dizziness
• αυτόν τον βήχα = this cough

This structure is extremely common in Modern Greek:

• αυτό το σπίτι = this house
• αυτή η γυναίκα = this woman
• αυτόν τον άνθρωπο = this person (masculine accusative)

English does not use both this and the, but Greek does.

Both are possible in writing, but τη is the shortened form of την before a consonant, and it is very common.

So:

• αυτή τη ζάλη
• αυτή την ζάλη

Both mean the same thing.

In everyday Modern Greek, the shorter form is very common before many consonants. You will also see the full form την kept in more careful writing, or when pronunciation makes it clearer.

Because βήχα is masculine accusative singular, and both the demonstrative and article must match it.

The noun is:

• nominative: ο βήχας = the cough
• accusative: τον βήχα = the cough as object

So:

• αυτός ο βήχας = this cough
• αυτόν τον βήχα = this cough as the object of the verb

Since αντέχω takes a direct object, Greek uses the accusative:

• αντέχω αυτόν τον βήχα = I can stand this cough
• δεν αντέχω αυτόν τον βήχα = I can’t stand this cough

Yes, mostly. Grammatical gender in Greek is something you usually have to learn with each noun.

Here:

• η ζάλη = dizziness → feminine
• ο βήχας = cough → masculine

That affects:

• the article
• the demonstrative
• adjectives
• some noun endings

So in the sentence we get:

• αυτή τη ζάλη because ζάλη is feminine
• αυτόν τον βήχα because βήχας is masculine

A good habit is to learn nouns with their article:

• η ζάλη
• ο βήχας

not just the bare noun alone.

Because they belong to different noun patterns, but both are in the accusative singular here.

• η ζάλη → accusative τη ζάλη
  Feminine nouns like this often look the same in nominative and accusative.
• ο βήχας → accusative τον βήχα
  Many masculine nouns in -ας lose the final -ς in the accusative singular.

So the difference is due to noun declension, not because they play different roles in the sentence. Both are direct objects of αντέχω.

Yes. That would still be correct:

• ... γιατί δεν αντέχω άλλο τη ζάλη και τον βήχα.

This would mean because I can't stand the dizziness and the cough any longer.

Using αυτή and αυτόν adds emphasis, something like:

• this dizziness
• this cough

It can make the complaint feel more immediate and vivid.

No, Greek word order is relatively flexible, though some orders sound more natural than others.

The original sentence is very natural:

• δεν αντέχω άλλο αυτή τη ζάλη και αυτόν τον βήχα

You might also hear variations such as:

• αυτή τη ζάλη και αυτόν τον βήχα δεν τα αντέχω άλλο
  More emphatic, with topicalization
• δεν αντέχω αυτή τη ζάλη και αυτόν τον βήχα άλλο
  Possible, though less neutral

The original version is a good standard model for learners.

You would usually say:

• Θέλω να γίνω καλύτερα σύντομα.

or more naturally in many situations:

• Θέλω να αισθανθώ καλύτερα σύντομα. = I want to feel better soon.

Difference:

• να γίνω καλά = to get well / recover
• να αισθανθώ καλύτερα = to feel better

So the original sentence suggests real recovery, not just a slight improvement.