(based on http://krdic.naver.com/detail.nhn?docid=13339800
1. enumerates multiple things.
- 이것은 감이며 저것은 사과이다 - This is a persimmon, and that is an apple.
- 남편은 친절하며 부인은 인정이 많다 - The husband is kind, and the wife caring and sweet.
2. describes two things happening at the same time (= ~면서).
- 그는 시인이며 교수이다 - He is a poet, and at the same time a professor.
- 그 집 아이는 공부도 잘하며 운동도 잘한다 - That family's kid is good at school work as well as sports.
Unlike ~며, ~고 has the sense of two or more things happening one after another, except in idioms like ~고 싶다.
(숙제를 하고 잤다 = I went to bed after doing my homework)
"indirect speech" you mentioned are the ~다며 and ~라며 forms (note 다 and 라), and means "saying that ...".
1. ~다며 = contraction of ~다고 하며 (= ~다고 하면서). Quotes a general verb/adjective-based phrase.
2. ~(이)라며 = contraction of ~(이)라고 하며 (= ~(이)라고 하면서). Quotes an ~이다(is)-based phrase.
(Another answer explains the difference between ~ㄴ다는, ~다는, and ~(이)라는 which is also relevant here - https://www.italki.com/question/385645
- 친구는 일이 있다며 [있다고 하면서] 먼저 일어섰다 - My friend rose first saying he had an errant to take care of.
- 동생은 자기가 1등이라며 [1등이라고 하면서] 뽐냈다 - My brother bragged saying that he got the first place.
So the quoting function comes from 다 and 라 added before ~며 (~며 itself only has the #2 sense above).
Such contracted 다/라 works with other conjugation forms too. We seem them in phrases like 행복하다고 한다, 슬프다며 울었다, 자기 잘못이라는 고백, 춥다면서 웅크렸다, etc.
Feel free to ask any further questions you might have.