I’m an American. Suppose I want to buy a summer home in Quebec. How is it it different than buying a house here? Do I need an paperwork beyond what a Canadian Citizen would need? How long can I stay there at one time without being a Canadian resident?
The answer to your last question is 6 months - same as the amount of time Canucks can stay in the US. I don’t know about the wrinkles of Americans buying up here, though. The fact that it is QC you’re looking at may complicate matters because they tend to take bureaucracy to another level there!
One thing Canadian snowbirds learn is that non-resident home owners can be subjected to any taxes that the municipality wants to levy on them - here’s your foreign ownership tax of 50%, for a made-up example. You’d probably want to do some research into buying foreign property before you buy anything.
I’m not sure entirely about this, but your taxation could be impacted, as well. In Canada our primary residences are not taxed as capital gains; other properties are, though. You might run into the same thing.
The major differences for an American buying a property in Canada:
The use of lawyers. The majority of real estate deals in Canada are done by lawyers. The realtor helps you fill out the offer (standard form, depending on province), and if it is accepted, the matter is in the hands of the buyer’s and seller’s lawyers. The deposit is done through the realtor, but title searches are done by the lawyer’s office, and the mortgage monies go through the lawyer’s trust account (i.e. escrow).
Knowing the difference between a Torrens title system and a deed registry system. Depending on province, there is one system, or a hybrid of the two. Western provinces have always used a Torrens system, but the eastern provinces haven’t. Ontario is slowly changing over, as properties change hands, but I cannot speak for Quebec. Still, you need to know exactly what you’re getting–your title may not give you everything you expect it to.
Title insurance. Related to the above. Under a Torrens system, you generally don’t need it for a residential sale or purchase: if prior title was clear title, and the Real Property Report indicates no irregularities, then yours will be too. Under a deed registry system, you may want to consider it
Citizenship doesn’t matter–as an American, you can buy as much land as you’d like. The amount of time that you can visit it, however, is left to Parliament and the CBSA.
NM, duplicate post.
There is not a uniform method in the US. It varies by state, and some are like what you describe in Canada.
But you forgot the biggest difference: In the US, you buy a house. In Canada, you buy a hoose.
OK, someone had to do it! 
In Quebec, though, I believe this is more typically done by a notary.
In Quebec, a Notary (notaire) is a lawyer; a specialised branch of the legal profession, but a lawyer.
I guess it depends how you define “lawyer.” Notaries must have a law degree but they are not members of the bar. However, I mentioned this because, practically, when you buy real estate in Quebec, the person that you will need to find will advertise himself as a notary (notaire) and not a lawyer (avocat).
Sure they’re lawyers. They have to have the standard three year law degree that the avocats get, then an additional fourth year, specializing in the areas of notarial law. They then must be admitted to the Chambre de Notaires, the equivalent to the Barreau, which is a member of the Federation of Law Societies of Canada. It’s a divided profession, with some similarities to the barrister/solicitor split in England, but I don’t think there’s any doubt that Notaires come within the general category of “lawyers.”
Not to turn this in a highjack, but this is all that I meant. Practically, it’s a different profession. For someone looking to enter the real estate market in Quebec, it’s useful to note that the people who will handle the legal paperwork are called and advertise themselves as “notaires” (notaries), not “avocat” (lawyer). As a matter of fact, I think there may be a bit of linguistic confusion, as I can find no French-language reference that describes “notaires” as “avocats”: they’re both “juristes”.