Past modals

[lcchang]

Dear teachers,
I would lile to ask you some big questions, and please be any help.
I read in my text book about the use of past modals for degrees of certainty and opinions and advice.
For degress of certainty, the book says:
Probability:
She must have left already.
She must not have wanted to come.
She couldn't have forgotten the party.
Possibility:
She may have forgotten our invitation.
She might have forgotten the time.
Her car could have broken down.
She may not have wanted to come.
She might have remembered the time.
My questions:
#What is the difference between probability and possibility?
#Why "must" and "could" are for probability but "may", "might", "could" are for possibilty?
#Does the past modals uesd here mean something happened in the past? If it is true, then why can "may" be used here since it is a present modal?
#Can we consider all the sentences above to be conditoinal sentences because they all look like in past perfect tense here?
For opinions and advice:
He could have left earlier.
He shouldn't have stayed so late.
I would have asked him to leave.
I wouldn't have stayed so late.
He could have been more considerate.
You could have reminded him of the time.
My questions:
#These sentences seem to be more like conditional sentences. Something did not come true but someone dreamed about it. Right?
#These sentences (opinions and advice) are a lot similar to the former ones (degrees of certainty), how do we tell them apart, or do they mean the same to you?
#Why there is no "may" to be used here?
Thank you for your time and I am sincerely looking forward to your advice.
LCChang