I agree that Canada would be merely a geographical term by the mid of the 19th century if annexed in 1812. Maybe not even that, or one of the North-Western states is named Canada (a bit analogueous to Louisiana). IMHO, the main catalyst of eradicating a distinct Canadian identity would be immigration. Nobody would decidedly migrate to Canada. They would go to America.
Even more so if Canada joins the US by its own free will during the American Revolution. By 1850, it would be "that colder region of America where most guys speak French". It would keep a shade of distinct cultural identity owing to the prevalence of French and Catholicism, but otherwise the Atlantic states would be merged with New England as a cultural and socio-economic region, Quebec would be a New England/Middle Atlantic hybrid, Ontario would be a part of the Midwest, the Prairie provinces would be indistinguishable from the Prairie/Rockies states, and Columbia from Washington and Oregon.
As it concerns naming and state border issues, my basic assumption is that OTL Quebec, Southern Ontario, and Northern Ontario would become three separate states. It's a coin's toss whether the former state gets named Canada or Quebec, but perhaps Canada is slightly more likely. It could easily go both ways. As it concerns the latter two states, in all likelihood, they shall have quite different names. In any case, America shall absolutely need an handy regional name for the row of states north of the St.Lawrence and the Great Lakes. PEI is absolutely going to stay a part of NS, far too sparsely populated to ever earn statehood. OTL New Brunswick may stay a part of NS or earn separate statehood much like Maine, but in any case it shall stay named Acadia. The prairie states and Columbia may or may not keep the OTL names, and the borders might be a bit different ITTL. Yukon and Alaska in all likelihood would be merged in one state. "Canada" is an obvious option, especially if Quebec keeps its OTL name. Otherwise, they would have to devise something else, e.g. "Upper North".
For that reason, you are right, Canadian secession is ASB. Mexican secession though...you never know, the annexation wouldn't be long ago. The US wouldhave to be an excellent occupator...
Quite true. However, they would be no friends of the CSA, owing to the slavery issue. My basic assumption is that in the wake of the secession, the Mexicans would try to stay neutral: they might still have lingering resentment for the Union because of relatively recent annexation, but they would rightfully fear that the Confederacy (or France) would swallow them and remold them in its image if it gets half a chance. My most likely assumption is that ITTL the Confederates or their French allies would overstep themselves with some aggressive move on Mexico, which would make the Mexicans throw their bets with the Union. The common struggle would largely bury American-Mexican lingering antagonism.
I also doubt that slavery would expand into Mexico, maybe even Texas would be spared the peculiar institution.
For geographical reasons, most US immigrants to largely empty Texas and Rio Grande would be Southern slaveholders, which is why I expect Texas to go the OTL way. ITTL I also expect the South to be bolder with such expansion, and hence the Rio Grande republic to follow the path of Texas. I wholly agree that there is no real possiblity that the South would ever manage to impose slavery on the popolous southern core of Mexico, regardless of whether ITTL it becomes an annexed US territory or a protectorate puppet.
Just like OTL, only more so, the vicious sectional struggle shall be about the relatively scarcely populated former northern Mexican territories, in addition to the OTL Lousiana Purchase and Mexican Cession, which shall surely become US territories and geography makes an open race between Northern freesoiler and Southern slaveholding settlers. It is likely that at the federal level, some kind of "popular sovreignty, let hte locasl decide" compromise over the issue of slavery in the territories would be reached. Like OTL, the South would soon realize that in an open contest, freesoiler settlers are going to win the day in most of the West, and that Mexicans were hostile to chattel slavery, much as they fancied peonage. Disillusionement about the possibility to replenish their ranks with many new states in the West would turn the South to secession. Since the above factors would make the sectional struggle go vicious faster, and the South's position would already be worsened by the presence of the Canadian free states (although ITTL I expect that the South would have previously strived to replenish its own ranks: expect the USA to acquire Cuba and Santo Domingo before the ARW, admit West and East Florida as separate states, Texas and Rio Grande to be split in several states), the ACW shall come 5-10 years faster. My reasoned guess is after the 1856 election, but 1850-52 is wholly possible.
To underline it...these annexations will weaken the "Cause" considerably. I hate to disappoint you, but maybe the whole Civil War gets cancelled due to that.
I rather doubt that the South would give up without a fight ITTL. They would be thoroughly persuaded that an independent Dixie-Caribbean slaveocracy would be viable thanks to the revenues of Cotton and Sugar, and that their boys would easily defeat the North in a fight thanks to their superior fighting spirit.
By the way, annexation of Mexico prior to the 1840s? I would say we are in wank-territory, despite the theoretical possibility. It would require an almost Neo-Spartan USA. With such a premature expansion, overstretch is just around the corner!
Many people make the mistake to assume that the relative US military weakness, as shown by the War of 1812, was the high mark of what the USA could do before the Mexican-American War. It ain't so, by any means. Such weakness was largely self-imposed by political choices, especially by the influence of the Jeffersonian Democrats, which largely gutted US military. Butterfly the Federalists into remaining the other major party, and more success than OTL. They had ambitious but wholly affordable plans for military expansion and readiness, which would have made the USA rather more formidable militarily than OTL in 1800-1846. Not to mention that owning Canada is going to make the USA significantly stronger.
On Party Politics, I still think that a 3rd or 4th party is viable if it can rely on a geographical stronghold. Look at the German Kaiserreich, despite a majority-voting-system it had several major parties.
Once Mexican voters get reconciled with their destiny in America (which I see basically happen by the Reconstruction, and done for good by the Gilded Age), there do not seem to be strong motivators for keeping a Mexican regional party. The social and economic interests of the Mexican voters can be better fulfilled by closing ranks with like-minded voters of other North American regions. The Canadien precedent makes a separate political identity for the Mexican voters out of religious and linguistic issues rather implausible. ITTL Catholic French-speaker constituencies would have gotten themselves wholly entrenched in the two-party system, and if they do so, why not Catholic Spanish-speakers ? As always, the wild card here is the South, and how it would deal with the Reconstruction ITTL, whether racial segregation still arises or is snuffed out in the crib. Butterflies arising from that and from Mexico's presence could make a three-party system viable, most likely made up of the Federalists/Republicans, the Democrats, and the Populists/Progressists.