I think so, but you can't apply for permanent residence if you came to the US to study. You have to first transition to a diffenrent visa status, usually H1-B.
However if, for example, the cap has been full since you graduated college...
Yes. They are subject to a lottery when te number of applications exceeds the number of available spots. This has been true for the last couple years, and last year was almost three times over.
I'm in the exact same situation. I wasn't selected for the H1-B lottery twice in a row, and have to leave after seven years in the US.
Oh well, I hear Vancouver is nice :)