Learn Hardcore Greek: Κάποια περιμένει έξω.

It’s the feminine singular indefinite pronoun meaning someone (female) or some woman. It stands on its own as the subject, without any noun after it.

Does this sentence specify the gender of the person waiting?

Yes. Κάποια indicates a female. For a male or gender-unknown generic you’d use κάποιος. For a non-human “something,” Greek typically uses κάτι.

Why isn’t there an article (like μια) before Κάποια?

Because κάποια is a pronoun, not a noun. It already carries the “some/someone” meaning, so no article is needed. With a noun, you could say κάποια γυναίκα (some/a certain woman).

What nuance does κάποια have compared to μια γυναίκα?

Κάποια (γυναίκα) = some/a certain woman (identity unknown or not specified; can sound slightly dismissive or distancing in some contexts).
Μια γυναίκα = a woman (plain indefiniteness, more neutral).

What tense/aspect is περιμένει, and why does it translate as “is waiting”?

Περιμένει is present tense, 3rd person singular of περιμένω. Greek present covers both English simple and progressive, so it can mean either waits or is waiting; context usually corresponds to the English progressive here.