It's better if you stay in OPT for about 6 months. Lots of money will be cut from your salary as tax once you are in the H1B. You can skip some of the taxes while in OPT. For H1B to be approved, it usually takes 3-6 months. I am almost exactly in your shoes besides that my OPT started in January and planning to apply for H1B during this summer.
And yes you are right that university does not fall in the quota but better to confirm once you are into the work.
Goodluck