F Visa

The processing time for the Canadian passport might be longer, so I suggest you check with the consulate. This situation appears unusual to me. In my understanding or recommendation, the law implies that once you take the oath of naturalization, it's the same as in the USA. I'm unsure about Canada's specific rules, but when you take the naturalization oath, you typically cease to be a citizen of any country except Canada. 

You may need to explore options to expedite your passport application, perhaps by paying emergency fees or any other required charges. I believe you should be able to enter the United States with a Canadian passport and an I-20 without needing a visa. Canadian citizens usually do not require a visa for entry.

If your green card application is family-based, such as through your spouse, and you have student status, it could potentially become a problem if you travel. In your case, I would suggest traveling only after you obtain either an H-1B visa stamping or Advanced Parole.

A. Certainly, there is absolutely no problem with that; a green card can be in process, and an H-1B can be filed.

B. No. The pending green card application should not hamper or impede the grant of an H-1B visa because the H-1B is a dual intent visa.

You can transfer an approved H-1B selected in the lottery to a new employer before October 1st.