As far as my knowledge is concerned SSN is not required for filing H1B if you don't already have it. Since people who come from India/Nepal with H1B visa directly don't have their SSN either.

Regarding OPT, I think you will be ok if you submit the receipt notice and the I-20 that shows you applied for OPT.

But the best thing is to consult with an attorney. Every company has their own attroney. So they might tell you best.

What I know is you can file H1B if you meet any degree (undergradaute or Masters) requirements while still being a full time student(F1). But the thing is that you can start working only from Oct1st once the H1B becomes effective.

Best of luck.