Can, could, and be able to

These auxiliary verbs are popularly used to talk about ability. They are quite analogous so that they sometimes are confused. It is a basic staff, so I wanna understand them clearly. 

• Can. This auxiliary verb is probably most frequent in talking about ability. For example: I can sing; I can do that exercise, etc. We usually use can to talk about general ability - the present simple. Therefore in other tenses as past or perfect we can use could or be able to. For example: I am so tired, I haven't been able to sleep last night; We stayed in a hotel near the beach. We could see a fantastic view of sea. 
• Could. This auxiliary can be used as the past form of can. Especially with the verbs like: see, taste, hear, remember, feel, understand. Could can talk about the permission in the past, for example: We were free in that house, we could do everything we wanted ( we were allowed to do everything we wanted).
  Another meaning of could is suggesting. We can use it to suggest or talk about possible now or in the future. For example: What do we have for dinner? We could have fish. Possibly we can say: we can have fish. In this case, with can the suggestion is more sure. I mean, maybe the speaker has bought some fish we she says: we can have fish. With could, maybe, speakers are standing in the front of a supermarket. One of them suggests that they can buy some fish for dinner.
  When we talk about actions which are not realistic, we use could. For example: I am utterly exhausted. I could sleep all week. For this aspect, we can compare can and could like this:
  This fish dish is so delicious, I can eat the whole dish. / This kind of fish is so delicious, I could eat it everyday.
• Could talks about ability in the past, but in general. If there is a particular situation, was/were able to or managed to. For example: Manchester United could beat any opponent when King Eric played for them. In FA final in 1995-1996 season, Liverpool played very well but he and his teammates managed to win. The fire spread very quickly so that everyone could get injured. Fortunately my friends were able to escape safely. 
• Couldn't, on the other hand, can be used in any situation with the meaning of failing to do something. For example: When I was a teenage, I used to like a girl. However, I was a coward in love so that I couldn't tell that I liked her. 
  Another meaning of couldn't is that the fact is impossible now. For example: Manchester United is 10 points less than Chelsea while the league has 3 rounds to the end. They couldn't win the tie this year.
  Everything is OK? Yeah, it couldn't be better [Big bang theory - Howard]
• Could have done, could have happened and couldn't have. All of them talks about the past. Could have done is the past of could do. For example. I am utterly exhausted. I could sleep all week./ I was utterly exhausted. I could have slept all week. 
  On the other hand, could have happened is about something possibly happened in the past but didn't happen. For example: Why were you angry with him although you didn't know if he had done it? You could have asked him first!