Getting a Chinese visa in the US as a Canadian citizen

In your case, I would contact your nearest Chinese consulate that services your area, and ask whether you can get a visa through them. Some consulates offer visa services to both citizens and permanent residents of the country they are in, and a TN1, although officially temporary, might be permanent enough for them to help you. In my experience the Chinese generally go "by the book" but are willing to bend the rules if you can talk to somebody with enough authority. In fact, this might be fine according to the Chinese consulate in Los Angeles:

(3) Proof of legal stay or residence status (applicable to those not applying for the visa in their country of citizenship)

If you are not applying for the visa in the country of your citizenship, you must provide the original and photocopy of your valid certificates or visa of stay, residence, employment or student status, or other valid certificates of legal staying provided by the relevant authorities of the country where you are currently staying.

Technically, you should be in possession of your passport (or at least have it available at short notice) whenever you are in the US (an immigration officer once described it to me as "except when you're swimming"). However, in practice this is unlikely to be a problem for you if you need to send your passport anywhere. Just don't do anything that might bring yourself to the attention of the authorities (same as usual, really).

As a Canadian citizen, you always have the right to return to Canada, but the burden of proof that you are in fact a Canadian citizen is on you. A passport is obviously the easiest way to prove this, but a birth certificate may be sufficient.