Those two structures are used for an assumption or a possibilities that includes an indefinite piece of information or a guess. For example, The team(팀) can be eliminated in the preliminaries.(팀이 예선탈락할 수가 있다.) If someone tell me that he or she met the Jesus Christ in the flesh on last Sunday, then I may say"Oh really? It might be true.( 그 것이 사실일 수도 있겠지.) If that is really true, you're going to go heaven soon. Congrats!" OR "It can't be true!( 그 것이 사실일 수가 없어!) Do you have any proof?"

Yeah, ㄹ 수 있다/없다 means I can / I can't, ㄹ 수도 있다/없다 I could / I couldn't. Here 수 means a way. In English, you'll just say "I can't sleep here", but in Korean you could say "여기서 못 자" or "여기서 잘 수가 없어". The first one is more common in everyday conversation. When you use ㄹ 수가 없다, it can be tricky to figure out which particle to use here. 여기서 잘 수가 없어 There's NO WAY for me to sleep here. 여기서 잘 수는 없어. There's no way for me to sleep here(although there may be some way for me to do something else, like eating or hanging out with my friends etc). So if you want to say "We can be winners", you can say "우리는 이길 수 있다" meaning there's a way for us to win. It would be 우리는 승자가 될 수 있다 if you want something that fits the English more literally, but Koreans love to use verbs. : ) Hope this helps.

As far as it is taught at the university, ㄹ 수 있다/없다 and ㄹ 수가 있다/없다 have the same meaning. Here '가' is like an ending for a noun