I'm not a native speaker, but I'll try to answer your questions!
-(으)ㄹ래요. I tend to think of this as "Shall we/I ___?" It could also mean "Do you want to __?". I think this grammar point is used more in a question form, not declarative, but it's still used both ways:
나: 우리는 뭐 먹을래? (What shall we eat?)
가: 흠...김밥요? (Hm...kimbap?)
나: 지금 뭐 해요? (What are you doing now?)
가: 늦어서 이제 잘래. (It's late so I shall/will go to sleep.)
-겠어요. This is just the basic way of saying something in the future. Like: 하겠어. However -겠어요 is usually used when you're using the high honorific form or when you say something seems like it would be good/fun/etc.
가: 네, 잘 먹겠습니다. (Yes, I'll eat good.)
나: 와~그 밥은 진짜 맛있겠다! (Wow, that food looks so good!)
-(으)ㄹ 께요. I think you mean ~(으)ㄹ게요. Sometimes it's written like that, since it's said like that.This is another future tense form however, this has a slight nuance that you're asking someone if it's okay.
나: (널 위해) 문을 열을게. (I'll open the door [for you] 'if that's okay'.)
가: (당신을 위해) 밥을 살게. (I'll buy food [for you] 'if that's okay with you'.)
다: 잘게. (I'm going to sleep now 'if that's okay'.)
- (으)ㄹ 거예요. Unlike ~(으)ㄹ게요, - (으)ㄹ 거예요 means you ARE going to do it, without the person's "permission". However remember it's just a slight nuance though, they can be interchangeable.
나: 내일 학교에 갈 거야. (I'm going to school tomorrow.)
가: 엉덩이 아파서 앉을 거예요. (My butt hurts so I'm going to sit down.)
I hope I answered your questions! ^^ If you have any questions, just ask~