Learn Hardcore Greek: Αυτή ξέρει ελληνικά πολύ καλά.

Yes. Greek is a pro‑drop language, so Ξέρει ελληνικά πολύ καλά is perfectly natural. Keeping Αυτή adds emphasis or contrast (She, specifically, knows Greek very well).

Does Αυτή mean she or this (woman) here?

In this sentence it means she (3rd‑person singular feminine). Αυτή can also be the demonstrative this (fem.). As a demonstrative before a noun you would say αυτή η γυναίκα (this woman). Without a noun, context decides between she and this one (f.).

Why is ελληνικά lowercase and in the plural?

In Greek, names of languages are not capitalized: ελληνικά, αγγλικά, γαλλικά.
ελληνικά is neuter plural and functions as a noun meaning (the) Greek (language). Historically it means Greek things; in modern usage it just means the language. It is grammatically plural but conceptually singular.

Should there be an article: τα ελληνικά?

After verbs like ξέρω/μιλάω, Greek usually omits the article: ξέρει ελληνικά. Adding the article is possible but changes the feel:

ξέρει τα ελληνικά can sound like speaking about the language as a subject or system.
You often see the article when specifying: Τα ελληνικά μου είναι καλά (My Greek is good).

Is the placement of πολύ καλά correct? Would Ξέρει πολύ καλά ελληνικά be better?

Both are acceptable. The most common neutral order is to put the manner adverb right after the verb: Ξέρει πολύ καλά ελληνικά. Placing it at the end (… ελληνικά πολύ καλά) is also fine and can sound a bit more emphatic on the adverb.

What’s the difference between ξέρω ελληνικά, μιλάω ελληνικά, γνωρίζω ελληνικά, and καταλαβαίνω ελληνικά?

ξέρω ελληνικά: I know Greek (general knowledge of the language).
μιλάω ελληνικά: I speak Greek (ability to speak).
καταλαβαίνω ελληνικά: I understand Greek (comprehension).
γνωρίζω ελληνικά: more formal; often used on résumés or in formal writing to mean you have knowledge of Greek. In everyday speech ξέρω/μιλάω are more usual.

How is ξέρω conjugated in the present?

εγώ ξέρω
εσύ ξέρεις
αυτός/αυτή/αυτό ξέρει
εμείς ξέρουμε
εσείς ξέρετε
αυτοί/αυτές/αυτά ξέρουν(ε)

How do I say she knew or she found out? Is there an aorist of ξέρω?

Modern Greek uses the imperfect for the past state of knowing and a different verb for learning:

ήξερε = she knew (was aware).
έμαθε = she learned/found out. There is no commonly used aorist of ξέρω. For present perfect, Greek prefers έχω μάθει (I have learned), not έχω ξέρει.

How do I pronounce each word?

Αυτή: af-TEE (the αυ becomes af before the voiceless τ).
ξέρει: KSEH-ree (single syllable onset ks).
ελληνικά: eh-lee-nee-KA (stress on the last syllable).
πολύ: po-LEE.
καλά: ka-LA.

What are the accent marks doing?

Modern Greek uses a single stress mark (τόνος) on words of two or more syllables to show the stressed vowel:

Αυτή (stress on -τή)
ξέρει (stress on ξέ-)
ελληνικά (stress on -κά)
πολύ (stress on -λύ)
καλά (stress on -λά)

Is καλά an adverb here? Why not καλή?

Yes, καλά is an adverb meaning well. It modifies the verb ξέρει. καλή is the feminine adjective good and would modify a feminine noun, not a verb. Greek often uses forms identical to neuter plural adjectives as adverbs (e.g., γρήγορα = quickly).

What’s the role of πολύ here? Does it change form?

Here πολύ is an adverb meaning very and is invariable; it does not change form. When πολύς/πολλή/πολύ means much/many as an adjective, it inflects. As an adverb before adjectives/adverbs/participles, it stays πολύ: πολύ καλά, πολύ καλός, πολύ γρήγορα.

Can the word order change for emphasis, like Ελληνικά ξέρει πολύ καλά?

Yes. Greek word order is flexible. Fronting Ελληνικά gives it contrastive focus: Ελληνικά ξέρει πολύ καλά, αλλά όχι Τουρκικά (She knows Greek very well, but not Turkish). Without a contrastive context, the neutral orders are preferred.

Why not αυτήν? When does the final -ν appear?

αυτή is nominative (subject) feminine singular, so no final -ν. The -ν appears in the accusative feminine singular (αυτήν) and is often retained before vowels and certain consonants. Example: Την είδα (I saw her). As a subject you use αυτή.

Is there any nuance added by starting with Αυτή instead of the verb?

Starting with the subject pronoun adds topicalization/emphasis, roughly She knows Greek very well (perhaps in contrast to others). The most neutral version would be Ξέρει πολύ καλά ελληνικά without the pronoun.