Learn Hardcore Greek: Πάλι δεν καταλαβαίνω τίποτα.

Yes. Greek uses negative concord: δεν pairs with items like τίποτα, κανείς, ποτέ. Δεν καταλαβαίνω τίποτα means I do not understand anything. Without τίποτα, Δεν καταλαβαίνω just means I do not understand (unspecified). Do not use Δεν καταλαβαίνω κάτι if you mean not anything; that means there is something I do not understand.

Can I use καθόλου instead of τίποτα?

They express different nuances:

Δεν καταλαβαίνω καθόλου = I do not understand at all (degree).
Δεν καταλαβαίνω τίποτα = I do not understand anything (quantity). For strong emphasis you can say Δεν καταλαβαίνω τίποτα απολύτως (absolutely nothing). Avoid stacking τίποτα καθόλου.

Can I move πάλι to another position?

Yes, with small shifts in emphasis:

Πάλι δεν καταλαβαίνω τίποτα: fronted, highlights repetition; often sounds exasperated.
Δεν καταλαβαίνω τίποτα πάλι: natural; gives an afterthought feel.
Δεν καταλαβαίνω πάλι τίποτα: possible but less common; ties again more closely to understand. Avoid Δεν πάλι καταλαβαίνω τίποτα.

What is the difference between πάλι and ξανά?

Both can mean again. Πάλι is extremely common and can carry a slightly annoyed tone; it can also mean then again (on the other hand), e.g., Πάλι, μπορεί να έχει δίκιο. Ξανά is neutral and often appears before the verb or as the prefix ξανα-:

Ξανά δεν καταλαβαίνω τίποτα (OK, neutral).
Past/repeated event: Δεν ξανακατάλαβα τίποτα (I did not understand anything again).

How do I say still rather than again?

Use ακόμα or ακόμη: Ακόμα δεν καταλαβαίνω τίποτα = I still do not understand anything. Πάλι does not mean still.

Why is there no subject pronoun like εγώ?

Greek is a pro‑drop language. The verb ending in καταλαβαίνω already encodes I. You add εγώ only for emphasis or contrast: Εγώ πάλι δεν καταλαβαίνω τίποτα.

How do I pronounce it?

Πάλι [ˈpali]
δεν [ðen]; before κ in καταλαβαίνω, many speakers nasalize to [ðeŋ]
καταλαβαίνω [katalaˈveno] (αι = [e], β = [v])
τίποτα [ˈtipota] Full phrase often sounds like: [ˈpali ðeŋ katalaˈveno ˈtipota].

Which tense is this, and how would I say it in the past?

It is present indicative (now or habitual). For a completed occasion in the past: Πάλι δεν κατάλαβα τίποτα = Again, I did not understand anything.

Is τίποτα always negative?

No. With negation it means nothing. In questions/conditionals it means anything or something:

Καταλαβαίνεις τίποτα; Do you understand anything?
Θέλεις τίποτα; Do you want anything? As a casual reply to thanks, Τίποτα! means It is nothing/You are welcome.

What about τίποτε?

Τίποτε is a stylistic variant of τίποτα. Same meaning; τίποτα is more common in everyday speech, τίποτε can sound more formal or regional.

Can I refer to a specific thing instead of nothing/anything?

Yes, use an object pronoun or a noun:

Πάλι δεν το καταλαβαίνω = Again, I do not understand it.
Πάλι δεν καταλαβαίνω την άσκηση = Again, I do not understand the exercise.

Is the sentence impolite?

It can sound frustrated. Softer options:

Συγγνώμη, δεν το καταλαβαίνω. Μπορείτε να το εξηγήσετε ξανά;
Δεν είμαι σίγουρος ότι το καταλαβαίνω.

Any colloquial intensifiers for the idea of nothing?

Informal: Δεν καταλαβαίνω γρι or Δεν καταλαβαίνω Χριστό = I do not understand a thing. Use with friends, not in formal contexts.

Should there be a comma after Πάλι?

No. Πάλι here is a simple adverb at the start; no comma is needed. The period at the end is correct.

Why not use μην instead of δεν?

Μην is the negative used with the subjunctive, imperatives, and certain non‑indicative structures. Δεν is the negator for the indicative. Since καταλαβαίνω is indicative present, δεν is required.