Learn Italian: Dopo essere sceso, ho comprato un gettone in quel chiosco. - After getting off, I bought a token at that kiosk.

The past infinitive uses the exact same helping verb that the verb normally uses in the past tense. Because scendere (to go down or get off) is a verb of movement, it takes essere. Therefore, we say dopo essere sceso rather than using avere.

If the speaker were a woman, would the first part of the sentence change?

Yes, it would become dopo essere scesa. Whenever you use essere as the helping verb, the past participle (in this case, sceso) must change its ending to match the gender and number of the subject. A female speaker would change the -o to an -a.

Why doesn't the second verb, ho comprato, also use essere?

Each verb in a sentence chooses its own helping verb based on its own rules. While scendere is a verb of movement that requires essere, comprare (to buy) takes a direct object (you buy a thing, like un gettone) and therefore uses the helping verb avere.

Why is it quel chiosco and not quello chiosco?