Your corp. must have register address in USA and that only happens when you are legal to run a business in USA as per FMCSA rep. But some companies like Autobahn Freight lines have US drivers in their fleet. Gave them a couple of loads last summer out of TX. they had US drivers on them who brought freight to their Romulus yard as per dispatcher for Canadian drivers.
As long as your tractors are IRP plated your equipment can legally work in the USA, regardless if the business is international, interstate, or intrastate commerce. Having a USDOT number is a condition of proper operating authority, not a condition of whether the equipment is legal or not. You do not need to be a U.S. registered corporation. However, you do need an EIN (employer identification number) from the IRS to pay your taxes.
As for drivers, Canadian drivers can work point to point in the U.S. under a very strict set of rules;
1) If the goods are of Canadian origin, and have not been processed, or enhanced in value in any way, a Canadian driver can pick up in the U.S. and deliver in the U.S. i.e. Canadian boxed widgets that are stored at a warehouse in Altoona, PA can be picked up and delivered in Denver, CO by a Canadian driver. We have already established the truck and trailer can do this legally. If Bubba Smith in Altoona so much as writes a number on the boxes, they are considered to have been processed and the rule does not apply.
2) A Canadian driver can switch trailers with another Canadian driver at a point in the U.S. and deliver to a point in the U.S. provided the goods are of Canadian origin,
and, it serves the exigencies of the hours of service rules.
3) Canadian drivers can re-position empty equipment between points in the U.S. provided they either a) enter and/or exit with a trailer, and b) the trailer is used in international commerce.
As to the OP's question ... " Can a Canadian based carrier use Canada plated trucks/trailers, BUT hire a US driver and do US to US work?"
The answer is unequivocally "Yes".
As an added bonus you can also hire North American Aboriginals, pay them in Canada, and avoid paying U.S. payroll taxes.