Might and may

All might and may talk about possibility. In this case, could is similar to may and might. For example: I has been calling Stephen 3 times but he didn't answer. He might/may be busy or he might/may be working or he might/may have forgot his phone at home. 

May and might are also used to talk about possibilities in the future. For instance: what will you do after finishing PhD? I don't know, I am considering, I might/may apply for some research jobs. Let's compare forms that talk about future suggestion or possibilities: will, be going to, could, might, may:

I will go to the gym today [possible]
I am going to the gym today [I have a plan to do it, it certainly happen]
I could go to the gym today. I feel so bored staying at home all day. [less sure]
I might/may go to gym today. I have a plenty of time doing nothing. [possible]
Don't call me at 8:00 pm tonight. I will be watching Manchester United vs Liverpool on TV. [ for sure]
Don't call me at 8:00 pm tonight. I might be watching Manchester United vs Liverpool on TV then. [possible].
For possible, only might can talk about unrealistic suggestion. For example: If I were you, I might resign from the company. [unrealistic I am not you].
Might as well is a special construction. It is used when there no reason not to do something. For instance: We have been waiting Mr Tom for so long. In fact, his secretary has said that he is going to be late in an hour and half. We might as well go to the cafe over there. There is nothing to do for us now. Or I have missed my bus to my office. The next one is in an hour. I might as well walk. An hour is so long to wait for the bus.

May not and might not are negative forms of may and might respectively. For example: He doesn't answer my calls. He might not hear the rings. 

Might have done and may have done are past forms of them. For example: I lost my keys. I have been looking them for hours at home. I might have left them at my office. 

Might not have done and may not have done are negative. Let's compare could not have done and might/may not have done:

Tom couldn't have seen you. Otherwise he would say hello to you. (it was not possible that Tom saw you without saying hello to you)

Tom might/may not have seen you. He walked through so quickly.