Learn Hardcore Greek: Αυτή δεν θέλει καφέ.

You can drop it. Greek is a pro‑drop language, so Δεν θέλει καφέ is fully natural and usually preferred when context already tells us who we’re talking about. Using Αυτή adds emphasis/contrast or clarifies gender when needed: e.g., “She (as opposed to someone else) doesn’t want coffee.”

Does αυτή mean “she” or “this (woman)”?

Both are possible, depending on context:

As a personal pronoun: Αυτή δεν θέλει καφέ. = She doesn’t want coffee.
As a demonstrative “this [fem]”: Αυτή η γυναίκα δεν θέλει καφέ. = This woman doesn’t want coffee. If you use αυτή alone and stress it, it can also mean “this one (fem)” in contrastive contexts.

Why is the negative δεν used here instead of μη(ν)?

Use δεν to negate statements and questions in the indicative (the default mood): Δεν θέλει καφέ. Use μη(ν) to negate:

imperatives: Μην πίνεις καφέ. (Don’t drink coffee.)
subjunctives/“να”-clauses when the negation targets the subordinate action: Θέλει να μην πιει καφέ. (She wants not to drink coffee.) In your sentence we’re negating “want,” so δεν is correct: Δεν θέλει να πιει καφέ. (She doesn’t want to drink coffee.)

Where does δεν go in the sentence?