What did the school's DSO say?
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.
Discuss OPT with the DSO. As to your H1B question, yes, any company can file an H1B petition for you requesting validity for 1.5 years. The requested start date can be immediate, as the petition will not be subject to the visa cap.
I agree with my colleague as to you being correct on the H-1B, and needing to check with the DSO as to the availability/length of OPT.
You would also want to speak with an attorney about your timetable for at least starting a green card case with a new, post-graduation employer to be able to preserve your ability to remain on H-1B beyond the initial six years.