Learn Croatian: Možete li staviti plombu bez anestezije? - Can you put a filling without anesthesia?

You are asking the dentist to complete the single, specific action of placing a filling in your tooth. When a modal verb like možete (can) refers to a single, completed event rather than an ongoing or repeated process, Croatian requires the perfective aspect.

In English, we usually say 'do a filling' or 'give me a filling'. Is staviti the standard verb to use with plomba?

Yes, in Croatian, the standard collocation for dental fillings is staviti plombu (literally 'to put a filling'). You shouldn't use literal translations of English phrasing like napraviti (to do/make) or dati (to give) in this context.

Why does the word anestezija change to anestezije at the end of the sentence?

It changes because the preposition bez (without) always requires the noun that follows it to take the genitive case. For feminine nouns ending in -a like anestezija, the genitive singular ending is -e.

Would I say Možeš li instead of Možete li if I am speaking to a single dentist?

You wouldn't usually. Možete li uses the polite Vi form (formal 'you'). In Croatia, it is standard and expected to use the formal register when speaking to medical professionals, so you use the plural verb form even when addressing just one doctor.