Response to your first question is "Yes". As long as you visa is not expired and the same with the I-20. You can travel as you wish. But there is a catch, As per your comments you have stated that you are about to complete your studies, now if you travel (assuming your visa  or I-20 is not expired) there could be a problem in the port of entry as they might ask you, what are you doing here after completing your studies? Now, lets say you applied for your Work authorization document and you travel while in OPT status, your OPT card wont be valid as it mentions in bold letters i think in the card itself, "Not valid after re-entry" or something like that. You would probably like to consult your international adivisor who could give you an exact answer.

If I were you i would keep things simple and be on the safe side. I'd wait till change my status. Preferebly H1B in your case and then travel which is much more safer. But to put it in a nut-shell you can travel despite which status your are in as long as you have a valid visa.

Good luck,

YekloYatri