1. YOu can use whatever you want, there's no requirment to use SIN, landlords will often want to do a credit check or get larger deposit. Obviously credit check hard for those who are new to the country. Perhaps others have suggestions. There are also firms and other ads for places that rent apartments or houses for longer or shorter terms; one year is considered a normal 'lease' but there are those who'd like to rent for 2-6 months too.
2. No, it's a jumble of different IDs in provinces. Health card, driver's license or Ontario photo card (only used for those who do not have driver's licenses), whatever. The truth is in Canada most places don't ask for id often, health card or dirver's license will be enough 90% of the time. You can of course use your foreign passport - but for things where you need to show residence in Canada, not great. (You can use PR card but we prefer not to carry it around). Oh, of course, increasingly lots of places are fine with pictures on phone (except for health card to receive health services).
3. Either is fine, only after landing. Online it takes a few weeks, service canada is right away.