我没做作业。 --> I did not do my homework. (If you want to say "I have not done my homework, I think 我还没做作业is better ...)
我不能做作业。 --> I cannot do my homework (or, I am unable to do my homework).
我不做作业。 --> (I don't do homework (that is, I generally don't do my homework)) OR (I won't do my homework)
我不可以做作业。--> I am not allowed to do my homework.
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In the examples you gave, 没 negates a sentence in the past, and 不 negates a sentence in the present (or future).
我没去 I did not go
我不去 I do not go, I am not going, or I will not go
不能 means cannot in the sense of unable to, and 不可以 means "not allowed to" or "not permitted to"
我不能去 I cannot go (I am unable to go)
我不可以去 I am not allowed to go
However, note that 没有 does not mean "did not have," but "do not have."