Everyone rushed to get on the elevator.
(Is 'to get-' the object of the verb 'rush'? Is 'rushed' a transitive verb taking an object here? Or Is 'to get...' an adverbial phrase modifying 'rushed'?))
http://grammar.ccc.commnet.edu/grammar/phrases.htm#infinitive
To get = infinitive
To get on the elevator = infinitive phrase
Yes, in this sentence, it's being used adverbially.