"저는 먹으면서 노래합니다."
Translation:I sing while I eat.
Singing is the main verb there, being the last one and with the -합니다 ending. Eat is followed by -으면서, therefore an auxiliary verb to express when the main verb happens.
Depends. In this sentence, it'd mean the same thing in English. Otherwise, it's definitely not "when". 면서 is used when two things are going on at the same time.