Learn Hardcore Vietnamese: Lan muốn gặp bác sĩ ở bệnh viện lớn. - Lan wants to see a doctor at the big hospital.

muốn means “to want.” In Vietnamese, when you express desire you use muốn + verb phrase. Here muốn gặp literally means “want (to) meet.” The structure is:
Subject + muốn + Verb (+ Object).

Why is gặp used here to mean “meet”? What’s the difference between gặp and thăm?

gặp = “to meet” or “to see (someone).” It emphasizes the act of encountering or meeting someone.
thăm = “to visit,” usually with a nuance of spending time with or checking in on someone (e.g., visiting a sick friend: thăm bạn ốm).
In your sentence, Lan isn’t “visiting” the doctor socially—she specifically wants to see or meet the doctor—so gặp bác sĩ is the natural choice.

Why isn’t there an article before bác sĩ, and how would I say “a doctor” or “the doctor” in Vietnamese?

Vietnamese has no direct equivalents of English articles (a, the). Nouns stand alone.

To say “a doctor,” you can add một (meaning “one”): một bác sĩ.
To say “the doctor,” Vietnamese often uses context or demonstratives: bác sĩ đó (“that doctor”) or vị bác sĩ (a polite classifier).
In your sentence, bác sĩ is indefinite and generic, so no article or classifier is needed.

What role does ở play in ở bệnh viện lớn, and how does it correspond to English prepositions?

ở is the “location” preposition, equivalent to English at or in. It marks where an action happens.
So gặp bác sĩ ở bệnh viện lớn = “meet the doctor at the big hospital.”
Location phrases in Vietnamese typically come after the verb (and object, if any).

Why is lớn placed after bệnh viện, and is this always how adjectives work in Vietnamese?

Yes—Vietnamese adjectives follow the nouns they modify.

bệnh viện lớn = “big hospital” (literally “hospital big”).
This word order (Noun + Adjective) is consistent: e.g., nhà đẹp (“beautiful house”), cây cao (“tall tree”).

How would I turn this statement into a yes/no question in Vietnamese?

Insert có after the subject and add không at the end:
Lan có muốn gặp bác sĩ ở bệnh viện lớn không?
This literally reads “Lan [does] want to meet the doctor at the big hospital or not?” and is the standard yes/no question form.