Note that there is also a word 조용히 which is perfectly (or almost perfectly) interchangable with 조용하게. Both mean 'quietly' and are derivered from 조용하다 - 'silent'
Does the ending (ㅡ)면서 have the connotation of "while"?
Yes, but only the kind of "while" meaning two things happening at once.
(1st action)-면서 (2nd action). = While (1st action), (2nd action).
"The man quietly sings while listening to the radio" was marked as incorrect.
"The man quietly sings while listening to the radio." is now accepted.
"the man listens to the radio while singing quietly" is wrong?
Because of the position of 들으면서 in the sentence, it has to be "while listening to the radio"
"While listening to the radio, the man quietly sings." the exact same meaning just in a slightly different arrangement...it's not wrong :(
"While listening to the radio, the man quietly sings." is now accepted.
The man quietly sings as he listens to the radio
why do we use 조용하게 here and not 조용히? what is the difference between the two?
Almost all -하다 verbs can be turned into an adverb by either changing it to -히 or 하게. There's pretty much no difference, except one is almost always more common than the other. 조용히 is used much more than 조용하게 but they mean the same.