I am on F1 visa in the US (Russian passport). Can I rent a car in Canada if I have a valid US-issued driver's license? Can I be listed on the car's insurance as a driver? I do not have international driver's licence. Not sure if it is relevant here, but my F1 is expired.
Yes you can.
I recently rented a car (as the only driver) in Canada (Montreal) with my US-issued license and I am not a US permanent resident (H-1b visa - expiring in a few months).
The car rental company doesn't need to see your passport or know your US resident status in order to rent a car to you - as long as you have a license which they recognize and either buy their insurance or provide your own, they're happy.