Sonora, Chihuahua, etc. had more people in it than New Mexico or California, but were still very empty. They are easily capable of being annexed to the US. The next tier of states, like Sinaloa, Durango, Zacatecas, are possibly doable but a bit more difficult. Here lies a much greater potential for warfare against the natives. Anything further south is just begging for trouble, except for perhaps Yucatan, which had a prominent faction which wanted annexation from the US because of difficulties against the local Mayans. But that in of itself is an issue--first, the US has to raise an army to go to the jungles to kill a bunch of Maya in what's no doubt the hardest Indian War they've faced up to that point, and second, it's highly likely US policy will alienate the people of the Yucatan before long.
The one big thing this will require is the United States raising a much, much bigger army than OTL, both pre-Civil War and post-Civil War. Most of it will be kept garrisoning Mexico. This will cost a significant amount of resources from the US. Further, the US will have to respond to local labor demands in ways the Mexican government could safely ignore (through bullets). This means they can't exploit Mexico to the degree someone like Porfirio Diaz did.
Something like Oaxaca or elsewhere in southern Mexico could easily end up being like Mississippi, except worse. Sure, Mississippi is better off than Oaxaca, but the US only has one Mississippi--here, you want multiple Mississippis, and you give a racial incentive to ignore the place. But Mexico, if annexed, could be brought up to Puerto Rico standards--best in Latin America, worse than the rest of the country. It might end up a drag on the rest of the country, and probably highly controversial, since the senators from Mexican states will be a huge bloc, as will their inevitably high amount of representatives. The alternative to that is keeping most, if not all, of it as a territory, and thus no voting rights, which might run afoul of international law and opinion once decolonisation as a movement takes off.
Mexicans may be divided and very messed up people at this point, however, faced with the consistent discrimination of a new government that sees them as "half Indian savages" and "mongrelized due to race mixing", who would only recognize a small fraction of them as citizens, not to mention the South inevitably attempting to force it's "peculiar institution" down their throats (and likely with the bulk of them on the receiving end), expect the bulk of the population to be in a permanent state of rebellion. Once the South seceedes, expect the Civil War to get MESSY...
Would the South actually enslave Mexicans? Even if they wanted to, wouldn't US law carry the day to make sure slavery was essentially limited to blacks, since Mexicans--even black Mexicans--would be free people? Unless they can succeed in getting something passed which allows people to sell themselves or their family into slavery, which I believe would be popular amongst some plantation owners (and would also apply to poor whites).
If the CSA tries to claim Mexico, they'll end up fighting a huge Mexican insurrection which will be aiming to restore the Mexican Republic. They won't be able to keep much.