Jintai C. answered 08/26/19
15+ years experience to teach Korean
First of all, 된, 되게, and 되다 are all different meaning and usage. Even all use 된, but the meaning is quite different: same shape, many different meaning and usage.
된 밥 : 된 means "Less moisture, dryness, not "hard-boiled".
Therefore, there is no word like "된 일" (X)
된장 : 된 means "real thickness", 장 means fermentation food
된장 is one of fermentation food and one of thickness fermentation food. that is why people use as 된.
되게 춥다: 되게 means in here is "very, extremely " and it is "Adverb.
춥다 is not verb. it is "Adjective. Some Korean Adjective look like verb, "ending with 다", but they are not verb.
되다 : verb: become, to be,
therefore, 된, 되다, 되게 have similar shape, 되 included, however, there usage is different, like a adjective, adverb, or verb and the meaning is all different.