This is somewhat of a broad topic, so we may bleed a bit into post-1900 territory here.
- If there is no War of 1812 (and no American invasion), do you think that any of the British North American colonies could/would have eventually joined the Union?
- Could Quebec ever successfully be a U.S. state?
- Could the U.S. have ended up with British Columbia, and if so, what would happen to the East?
- Under what conditions (if any) would Newfoundland have seriously considered U.S. statehood?
- When do you believe is the latest that an American invasion of Canada (and/or war against the U.K.) is a remotely plausible scenario?
- What level of civil resistance are we going to see if the U.K. "abandons" Canada to U.S. annexation/occupation?
- What sort of regional identity would an American Canada have? (Dixie 2: The Great White Boogaloo?)
- What would a rump/puppet Canada look like if the U.S. decided not to outright annex it?
I do realize that answers to some of these questions would vary greatly depending on when the events took place.
That said, I wanted to keep it somewhat broad to discuss a wider variety of possible scenarios.
1) Debatable - even post-1812, there were still some annexation movements. The idea of a separate identity for British North Americans would be much weaker though, as would the push west, and the British thinking they'd need to prop up BNA against the USA. The latter part is especially important - the more time given, the less of a chance they would have to do so, especially outside of Upper and Lower Canada.
2) Absolutely, perhaps much more successfully than it has been as a Canadian province. The US Federal system work would just fine at allowing Quebec to maintain several of its peculiarities - we even have a similar road map with Louisiana. I do however, think that Quebec would be broken into smaller states though - which given it gives Quebec a bigger voice in the Senate, might actually play out well with the Quebeci people.
3) Absolutely, and there are at least a dozen ways to do it, with PODs ranging from the Astor expeditions in the 1810s to settlement of claims between the USA and UK post-American Civil War. Impact on the USA would be small, but noticeable - we get probably two or three states out of it, and Alaska would be connected to the US mainland. The latter part may also come into play during negotiations over the border with the Yukon over the OTL Alaska panhandle - with the US in possession of OTL British Columbia, they may be able to push for even more land than OTL to allow a better land connection with the rest of the USA. If the USA plays its cards right, we could see the border drawn along the Yukon River watershed or extended along the Continental Divide.
The impact on Canada though would be profound - you can't have Canada as we know it without a Pacific coastline, and the reduced Canada would be much more centered on Ontario and the St. Lawrence valley. Which could have some interesting impacts on Western Canada, which even in OTL felt that Ontario ignored them. Without the need to link up with British Columbia and the Pacific, that would be greater than OTL, possibly resulting in the Prairie provinces either forming a separate Dominion or seeking to join the US.
If the USA takes everything west of the Continental Divide, the usual thing done here is one of two things, both fairly plausible - we get a weaker Canada with some greater regional tensions, or Britain might decide to forgo creating one Dominion, opting instead to create several. Rather than Canada, we could see a Dominion of Quebec, a Dominion of Ontario, and a Dominion of the Maritimes, and Newfoundland and the Prairies with rolled into one of them, sold to the USA, or in the case of Newfoundland, maybe even integrated into the UK itself.
4) This frankly depends on a number of factors - the big one being the time period. Just have the Brits and/or Canadians offer Newfoundland a raw deal, and the US offer a better one, and you have it.
5) 1890s, barring the very unlikely idea of the USA fighting for the Central Powers. The USA and Britain very nearly came to blows over British claims in Venezuela during the 1890s. Like most possible wars between the USA and Britian between 1848 and 1900, the later the war is, the better the USA will do, but by and large, the USA will overrun Canada with little issue, while the Royal Navy will harass the coast and prevent the US from getting ideas outside of North America. Only question is how much of Canada the US will take at the peace tables.
6) There isn't much Canada can do against the US should Britain abandon them to their fate - there is a reason why every defense plan for Canada pretty much consists of "hold as best as we can until the British can come and save our bacon".
The USA had the industry, the military, the resources, and most importantly, the population to completely subsume Canada. What resistance there is while be both fleeting and futile.
7) Depends on the part of Canada, especially since some regions of Canada might be less inherently "Canadian" than others. Other parts of Canada might group off culturally with similar parts of the USA - Columbia would fit right in alongside Washington and Oregon in the Pacific Northwest, while the Prairies might end up the northern reach of the Midwest or Plains states. The Maritimes have ties to New England that date back to colonial times, and they would enjoy a friendly rivalry with Massachusetts over which batch of plastic paddies can out Irish the Irish.
Everything west of Manitoba has the added benefit that it could probably be flooded with American settlers and immigrants.
That leaves Quebec and Ontario - Quebec has its own regional identity, which as pointed out before, would probably exist as a US State without much protest.
Which leaves Ontario, the Canadian heartland. Best case scenario, it could be a quirky regional identity like Texas. Worst case scenario, the Maple Leaf flag becomes the frostier cousin of Stars and Bars. If the latter, Northern Ontario probably gets split off, settled and integrated.
8) The US would absolutely take everything west of the Continental Divide, and as much of the Prairie provinces as it could. Anything that doesn't join the US will be broken into smaller, more easily influenced republics.
It would also try to see if it could get the Maritimes to join the US, and if not, spin it off as a separate republic. Relations are probably very warm with the USA, and the nation is known abroad for its fishing fleets, its merchant marine, and its distinctive dialect of Gaelic.
Which once again, leaves us Quebec and Ontario. If the Maritimes joined the US, a similar offer may be made to Quebec, otherwise, they'll be spun off as their own Republic. Very cordial relations with the USA if it goes independent, with whom it enjoys a brisk trade in everything from lumber to hydro power, proudly francophone, and it probably follows France's lead as much as it follows the USA's. Other national pastimes include poutine, and taunting the rump Canadians across the border.
Which again, leaves us with Ontario, depending on where between the Rocky Mountains and Lake Huron the final border with the USA is eventually drawn. Even if its just Southern Ontario, rump Canada is somewhat viable as an independent nation - to this day, its where 40% of Canada lives. Needless to say, the farther west that eventual the US border is, the more independent minded and self-reliant this rump Canada will be, but even in the best case scenario for rump Canada, the USA will loom over it like a colossus. If OTL Canada is touchy about its distinctiveness from the United States, you can bet your bottom dollar that this rump Canada will be even more so. Which could manifest itself in some interesting ways over the years. I doubt it will grow to bitterness, at least not to the point of hostility - with the USA on at least two sides of rump Canada, and friendly pro-American republics on the other side of it, when the chips are down, rump Canada jumps when the USA says "frog".