1. 밤 means "night" and 밤새 means "the night time" or "overnight", but I don't know what 새 means in this word. I know that 새 by itself can mean "new", but I don't know if that is its usage here.
=> 새 means 'an opening' or 'an interval', originally 사이.
So, 밤새 means 밤사이, the night time.
2. Can 새 be added to other periods of time? 주말새 = all weekend?
=> Yes, you are right. We usually do not use 주말새, but you can find 주말새 in some newspaper articles. eg. 프랑스서 주말새 잇단 화재… 8명 사망.
Also, you can use 한달새, 이틀새 etc.
3. 밤새도록 means "all night long" or "overnight". I know the meaning of 도록 but I think it can only be attached to verbs, right? Does it use the meaning "to the degree or extent" here?
'밤새'와 '밤새도록', 어떻게 뜻이 달라요?
=> 밤새도록 means 밤새다, literally '~ pass the night until the sun rises or the dawn' however, you are right, meaning 'all night long' and
밤새, 'the night time'. However, both words are sometimes exchangeable, 밤새도록 일했다. 밤새 일했다.
밤새우다 meaning, 'to stay up all night', 밤새워 일했다.
I passed a sleepless night.
뜬 눈으로 밤새웠다.
밤새 한잠도 못잤다.
밤새도록 한잠 자지 못했다.