YES

Yes if you can make the work course related. I did summer intern in a company during summer break during my 2 year MBA program. It is a different type of OPT, won't impact your 1 year OPT time after graduation. At that time (1998), I only needed to get the justifiaction letter from the professor and got the signature from the school International Student Office director on my I-20 before I started working. Not sure whether the process changed or not. Good luck