1.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?
Probably not. They would be more open to American immigration (but there's little evidence this would lead to much clamor for annexation) which would probably increase the population, but the Maritimes all felt a deep cultural/economic tie to Britain, what became Ontario had a proud Loyalist tradition which influenced it for over a century, and Quebec is pretty much a non starter.
2.Could Quebec ever successfully be a U.S. state?
Contrary to popular opinion I would argue no. The Louisiana argument has been floated, but can be challenged by the Canadien apathy to republican institutions and intense fear of being absorbed by a larger English population. Quebec has been politically restive ever since 1867, and that would be just as bad in this scenario, even with token protections Quebec would be hard core in the states rights camp and ultimately be a near eternal headache and would fiercely resist integration. Basically from 1775 on the chances of Quebec willingly being absorbed into the United States are nil.
Could the U.S. have ended up with British Columbia, and if so, what would happen to the East?
Possibly. Effectively until 1871 and the agreement to construct a railroad there was not much BC had in common with Canada save being a British colony. There wasn't much annexation sentiment OTL, but if you had more American settlers coming in the 1850s and 1860s, and getting integrated with the social elite then I see it as a distinct possibility a more active annexation movement could take hold. Heck, even delaying the Alaska purchase a while might remove the feeling of being surrounded and let them be more amiable to the idea.
Under what conditions (if any) would Newfoundland have seriously considered U.S. statehood?
As was pointed out earlier, its basically only if the Maritimes are already American. Newfoundland was dragged kicking and screaming into Canada as it was, and they only joined because the entire Dominion went broke and Britain basically said "Ok, time to join another country now..."
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?
1862 is the last time I see it being remotely possible. Others have mentioned a crisis over the Alabama Claims, but I'm skeptical a nation emerging from four years of civil war with a South to reabsorb and millions of freed slaves to deal with would be positively eager to go a round with Great Britain. Circumstances might push towards it, but I really can't see Andrew Johnson or US Grant being really eager to find a military solution.
I say 1862 because the much discussed Trent scenario is the closest either side came to at least severing diplomatic relations or open hostilities, and Britain was actively gearing up for war. After that she had good reasons to not desire to take on a fully mobilized America, and there was more to be earned via negotiation rather than gunboat diplomacy. Even the 1895 Venezuela Crisis was really only so much posturing.
What level of civil resistance are we going to see if the U.K. "abandons" Canada to U.S. annexation/occupation?
Depends on how its "abandoned" really.
What sort of regional identity would an American Canada have? (Dixie 2: The Great White Boogaloo?)
Well as previously mentioned Quebec is probably just as troublesome as it ever was, and scoffs openly at most of its neighbors, alt-Ontario probably adopts a very New England attitude, which means it largely shifts with the times and liberal mores of the age. The Great Plains probably add up to the rural conservative wild west culture that developed OTL in the American West, and BC is Oregon writ small (or large geographically speaking).
In interesting terms, if the whole slavery debate goes on track you might see slavery further north as the compromises become more difficult, and if you get secession some time in the 1850s-60s, you can bet your britches Quebec probably hops on that train in an effort to kick off the objectionable parts of the US system.
What would a rump/puppet Canada look like if the U.S. decided not to outright annex it?
Probably (depending on what was lost) and independent Republic of Quebec, and a Maritime Union which is tied closely to Britain. If Ontario still exists, its a very fragile province which looks West to the plains with envy.