A Canadian immigrant doesn't enjoy any special advantages over others. However, once you attain Canadian citizenship, numerous new opportunities become available. For instance, you become eligible for Treaty Trader and Treaty Investor visas like E-1 and E-2, as well as TN visas. Additionally, for most visa categories, you don't actually require a visa stamp. As a Canadian citizen, you can enter the United States without a visa, except for E visas and K visas. Visas such as F-1 and tourist visas typically aren't necessary; you can simply arrive at the border or airport with your Canadian passport. That's how the process works. It's important to note that being a permanent resident of Canada doesn't provide any advantage in the United States.

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.

Your employer is correct.

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.