Learn Hardcore Polish: Ona była tak zdenerwowana, że prawie zapomniała, po co tu przyszła. - She was so nervous that she almost forgot what she had come here for.

Polish does not require subject pronouns, because the verb ending usually shows the person.
You could say simply Była tak zdenerwowana, że… and it would still be correct and natural.

Using ona adds a bit of emphasis or clarity, for example:

when you contrast people (Ona była zdenerwowana, a on był spokojny.)
when you introduce or re‑focus on her in a story.

So ona is optional here, not grammatically necessary.

What exactly does the structure tak … że mean in this sentence?

Tak … że is a very common “so … that” construction expressing a degree that leads to a result.

Ona była tak zdenerwowana, że…
= She was so nervous that…

Compare:

bardzo zdenerwowana – very nervous (just degree)
tak zdenerwowana, że prawie zapomniała – so nervous that she almost forgot (degree + consequence)

Don’t confuse tak … że (two words, with a comma) with:

także – also, too
tak że used as a connector meaning so, therefore (e.g. Tak że wyszliśmy wcześniej.)

What form is zdenerwowana? Why is it not a normal verb?

Zdenerwowana is an adjective (historically a participle), in the form:

gender: feminine
number: singular
case: nominative

It agrees with ona:

Ona była zdenerwowana. – She was nervous/upset.
On był zdenerwowany. – He was nervous/upset.
Oni byli zdenerwowani. – They (mixed / all male) were nervous.

Polish often uses być + adjective to express states:

była zmęczona – she was tired
była zadowolona – she was pleased

Could we say Ona się tak zdenerwowała, że… instead of Ona była tak zdenerwowana, że…? What’s the difference?

Yes, that’s also correct, but the focus changes slightly:

Ona była tak zdenerwowana, że…
Emphasizes her state: She was (in such a state of being) nervous that…
Ona się tak zdenerwowała, że…
Emphasizes the event / moment of getting upset: She got so upset that…

Both are natural; the original version paints her as already in a strong emotional state.

What does prawie mean here, and where can it go in the sentence?

Prawie means almost / nearly.

Here it modifies the verb:

prawie zapomniała – she almost forgot

It normally stands just before the word it modifies.
You can’t move it very far without changing meaning or sounding odd. For example:

Ona była tak zdenerwowana, że prawie zapomniała… – natural
Ona była prawie tak zdenerwowana, że zapomniała… – now prawie modifies tak zdenerwowana, meaning almost that nervous, which is different.

Closest English equivalent here: “She was so nervous that she almost forgot…”

Why is it po co and not dlaczego? Aren’t both “why”?

Both relate to “why”, but they’re not the same:

po co = for what purpose, what for
dlaczego = for what reason, why

In this sentence:

po co tu przyszła – what she came here for / for what purpose she came here

If we used dlaczego:

dlaczego tu przyszła – why she came here (asking about the reason, not the goal)

Here we care about the purpose of coming (e.g. to do something, to meet someone), so po co fits better.

How does tu differ from tutaj? Could we say po co tutaj przyszła?

Tu and tutaj both mean here, and in most contexts they are interchangeable.

po co tu przyszła
po co tutaj przyszła

Both are correct. Tu is a bit shorter and slightly more colloquial in feel; tutaj can sound a bit more emphatic or careful, but the difference is subtle. In everyday speech, tu is extremely common.

Why is the word order po co tu przyszła and not something like English “what for she came here”?

Po co tu przyszła is an indirect question (embedded clause) with normal Polish word order:

Question form: Po co tu przyszłaś? – Why/what for did you come here?
Embedded: …po co tu przyszła. – …what she came here for.

Polish does not add extra inversion in indirect questions like English does.
You can move tu:

po co przyszła tu – also possible, but po co tu przyszła sounds more natural and fluid here.

Why are there commas before że and po co?

Polish comma rules are quite strict:

Comma before że
Że introduces a subordinate clause, so you almost always put a comma:
- …tak zdenerwowana, że prawie zapomniała…
Comma before po co here
Po co tu przyszła is another subordinate (indirect question) clause, so it also takes a comma:
- …prawie zapomniała, po co tu przyszła.

So the sentence splits into three chunks:

main clause 1: Ona była tak zdenerwowana
clause 2 with że: że prawie zapomniała
clause 3 with po co: po co tu przyszła

What tense and aspect are była, zapomniała, and przyszła, and why these forms?

All three are past tense, 3rd person singular, feminine forms.

była – from być (to be), past, feminine: she was
zapomniała – from zapomnieć (perfective: to forget [once / completely]): she forgot
przyszła – from przyjść (perfective: to come/arrive): she came

Perfective verbs (zapomniała, przyszła) describe completed events in the past.
That fits the meaning: she (already) came, and she almost completely forgot the purpose. Using imperfective forms (zapominała, przychodziła) would sound wrong or at least very different in this context.