Translation:Was his daughter born in a hospital?
You are right. Same as in many languages, in german "Krankenhaus", in Swedish "Sjukhus" (sick-house). Because birth may be dangerous for both, many people do it in a special part of a hospital so that "just in case" help is already in place. Just if they become "malsana", "malsaniĝas", they are in the right "ejo" :-) .
Besides, "Malsanulejo" is a vey general word, including sanatories, hospilals and so on. The word "hospitalo", which has a narrower meaning, exists, but is not so often used, only when needed to distinguish. And also "hospitalo" includes the birth (giving) part.
Is fiakro (cab) an accepted word for taxi in English? In English, I believe that both taxi and cab are shortened forms of taximeter-cabriolet (also taxicab). I was wondering if (fiakro) also had the other meanings that cab has in English, such as the cab of a lorry (truck in American). Cab is short for cabin in this case.