Learn Arabic (Modern Standard): هل عندك وقت؟

Arabic doesn't have a direct verb for "to have". Instead, it uses preposition-like words (sometimes called pseudo-verbs) combined with attached pronouns. عند means "at" or "with", and ـكَ (-ka) means "you" (masculine). So, "at you is time" is how Arabic expresses "you have time".

How do I change this to ask a woman if she has time?

You change the pronunciation of the attached pronoun suffix at the end of the word. For a male, it is pronounced with a short "a" sound: عندكَ ('indaka). For a female, it becomes a short "i" sound: عندكِ ('indaki). Without vowel marks, both are written the exact same way: عندك.

Is there a word for "any" or "some" hidden in this sentence?

No, Arabic doesn't need an extra word here! The word وقت (waqt) simply means "time". Because it is indefinite (it doesn't have "the" attached to the front of it), it naturally translates to "some time" or "any time" in an English context like this.

Why does the word for "time" (وقت) come after عندك?

In Arabic "have" sentences, the possessor part (عندك, meaning "at you") almost always comes before the thing being possessed (وقت) whenever that thing is indefinite (meaning it is just "time" rather than "the time").