‘Stuck up’ could also be used to mean arrogant but the important aspect is that it suggests that you consider yourself to be very important and you look down on other people. There is another expression that conveys the same idea: give yourself airs and graces or walk around with your nose in the air.
‘Beside yourself’ suggests that you are out of control emotionally and can refer both to pain as well as joy. You can be beside yourself with grief about what has happened and you can be beside yourself with joy about what has happened.
As a footnote: when you say that someone is “stuck-up”, it often implies that there is no justification for his air of superiority. We may be willing to admit that an “arrogant” person is justified in his high estimate of himself; but not a “stuck-up” person.
No; the subject of the verb must coincide with the object of “beside”. Thus “I am beside myself…”, “he is beside himself…”, “she is beside herself…”, etc.