Learn Hardcore Greek: Η καριέρα της προχωράει γρήγορα, αλλά ακόμα προσπαθεί να έχει ισορροπία στη ζωή της. - Her career is moving forward quickly, but she is still trying to have balance in her life.

You cannot say η της καριέρα in modern Greek; that feels archaic or wrong.
If you want to emphasize the possessor, you can add a stressed form, e.g. η δική της καριέρα (her own career), but the basic position of της stays after the noun.

In the second part, …αλλά ακόμα προσπαθεί να έχει…, there is no word for she. How do we know the subject, and is it correct to leave it out?

Greek is a pro‑drop language: subject pronouns (I, you, he, she, etc.) are usually left out because the verb ending already shows the person and number.

προσπαθεί is 3rd person singular (he/she/it tries / is trying).
From context, we know it refers to the same woman whose career was mentioned, so it means she.

You could say αλλά αυτή ακόμα προσπαθεί… to mean but she still tries…, with extra emphasis on she (for contrast). In neutral, everyday speech, leaving out αυτή is more natural.

Why is προχωράει used here for something like “is progressing / is moving forward”? Does Greek have a separate present continuous tense?

Modern Greek does not have a special present continuous form like English. The present tense usually covers both:

προχωράει = she moves forward or she is moving forward / is progressing, depending on context.

If you really want to emphasize ongoingness, Greek can use other expressions (e.g. είναι σε εξέλιξη, “is in progress”), but for normal verbs like προχωράει, the simple present is enough for both simple and continuous meanings. Here, context makes it clear we mean an ongoing process: her career is progressing quickly.

I sometimes see προχωρά instead of προχωράει. What’s the difference?

There’s no real difference in meaning:

προχωράει
προχωρά

Both are standard 3rd‑person singular present forms of προχωράω (to go forward / progress).

The version in ‑άει is a bit longer and often sounds slightly more colloquial or “spoken”, while the ‑ά form is a bit shorter and somewhat more common in writing. In everyday speech, you will hear both, and they are interchangeable in almost all contexts.

Why is γρήγορα used, and not γρήγορη or γρήγορο? How does this adverb form work?

γρήγορος is an adjective:

masculine: γρήγορος
feminine: γρήγορη
neuter: γρήγορο

But in the sentence, γρήγορα is an adverb, modifying the verb προχωράει (how does it progress? quickly).

In Greek, many adverbs of manner are formed from adjectives in a form ending in ‑α (historically related to the neuter plural):

γρήγορα = quickly
άσχημα = badly
ωραία = nicely, beautifully

So:

προχωράει γρήγορα = it/she progresses quickly

You cannot say προχωράει γρήγορη here, because γρήγορη is an adjective and would need to describe a noun, not a verb.

What exactly does αλλά ακόμα mean here? Could we move ακόμα or replace it with something else?

αλλά = but
ακόμα = still / yet

Together in this sentence: …αλλά ακόμα προσπαθεί… = but she still tries / is still trying…

You can move ακόμα around the clause without changing the meaning much:

αλλά ακόμα προσπαθεί…
αλλά προσπαθεί ακόμα…

Both mean but she still tries…; the first has a tiny bit more emphasis on the “still” right after αλλά, but both are very natural.

You can also replace ακόμα with:

αλλά εξακολουθεί να προσπαθεί… (a bit more formal: but she continues to try…)
αλλά συνεχίζει να προσπαθεί… (but she continues to try…)

So αλλά ακόμα is a very common, neutral way to say but still.

Why do we need να in προσπαθεί να έχει? What is this να construction?

να is the marker that introduces the subjunctive (the “dependent” or “non‑finite” form), and it often corresponds to English to in structures like try to do, want to do, decide to do.

After verbs like:

προσπαθώ (I try)
θέλω (I want)
αποφασίζω (I decide)
πρέπει (must / have to)

Greek uses να + verb:

προσπαθεί να έχει = she tries to have
θέλει να έχει = she wants to have

You cannot say προσπαθεί έχει ισορροπία; that would be incorrect. The να is necessary to form this verb‑phrase structure.

Could we say προσπαθεί να κρατήσει ισορροπία or να βρει ισορροπία instead of να έχει ισορροπία? Is there any difference?

Yes, those are all possible, but they have slightly different nuances:

προσπαθεί να έχει ισορροπία
Emphasizes the state of having balance in her life, more generally and continuously.
προσπαθεί να κρατήσει / να διατηρήσει ισορροπία
Implies she already has some balance and is trying to keep/maintain it.
προσπαθεί να βρει ισορροπία
Implies she hasn’t found balance yet and is trying to find it.

The original να έχει ισορροπία is a neutral, general way to say she is trying to have (maintain) balance in her life, without specifying whether she is finding it for the first time or preserving it.

There is no article before ισορροπία. Why don’t we say να έχει μια ισορροπία?

With many abstract nouns that refer to a general quality or state (like ισορροπία = balance, υπομονή = patience, ελευθερία = freedom), Greek often drops the indefinite article when the meaning is general:

να έχει ισορροπία = to have (some) balance, to have balance in a general sense.

If you say να έχει μια ισορροπία, it is grammatically correct but sounds more like:

to have a (certain, particular kind of) balance
or
to have some sort of balance (slightly more specific or a bit “countable”).

In this context, where we mean balance in life as a general quality, να έχει ισορροπία is the most natural choice.

In στη ζωή της, what is στη, and why do we repeat της again? Could we drop it?

στη is the contraction of σε + τη(ν):

σε = in / at / on
τη(ν) = the (feminine singular accusative)

So:

στη ζωή = in (the) life

ζωή is feminine, so it takes τη(ν). Written together and pronounced, σε την → στη(ν).

Then we add the possessive:

στη ζωή της = in her life

Again we use the pattern article + noun + possessive pronoun: τη ζωή της.

We repeat της because otherwise στη ζωή would mean in life (in general), not necessarily in her life. The της makes it clearly personal.

Normally you cannot drop that της here; να έχει ισορροπία στη ζωή would be understood more as “to have balance in life (in general)”, not “in her life” in particular. Adding της specifies whose life we’re talking about.