i am not 100% sure but i am fairly positive that it will fall under STEM.

Since the university is not going to sponsor your green card and you will be working for them temporarily, i think you can work for them on ur OPT...and then look for for-profit companies who will apply for your h1b...as soon as you find one and on the right time to apply for your h1, move on...i guess that will be best...