6265147
I would imagine that the Confederacy would essentially have a blend of Hispanic and Dixie culture if they won the war and conquered Mexico and the American Mediterranean. I think that modern Confederate culture would be similar to our real-world Southern culture, with some Hispanic culture thrown in for good measure. I doubt it'd be a member of the UN, as what the UN stands for nowadays would be incompatible with Confederate values and beliefs. No clue on who their ambassador to the Union would be.

6265147
No, this is the most likely shape of a modern Confederacy:

With a shit industrial base, jingoistic diplomacy, aristocratic economy, massive economic and social time-bombs, and vastly superior neighbor with territorial claims on it, the Confederacy would have been lucky to lurch zombie-like to 1900 before some casus belli with the Union put it out of its misery.

Ignoring that, I'd also dispute the map itself. Arizona and "Sequoyah" were absolutely out of Confederate ability to conquer, effectively requiring a second war in order to annex. Even Kentucky and weird half-Missouri proved out of reach (and if the Union is thoroughly drubbed, why settle for half of Missouri?). Pinching Cuba off Spain seems possible, but wars of conquest against Mexico would have brought global intervention against the CSA, and seems vastly out of its character - in the historical Mexican-American war, Southerners shied from the idea of annexing Mexico due in large part to their identity as a white nation. I don't see this mentality changing with the white supremacists triumphant. If it did, they would bring in a new underclass into the nation and do more to promote a US-Mexico alliance than any number of overtures could hope. I do not think they would have a Hispanic culture, any more than America has one.

"Alternate history" means you can say whatever, I guess, but I do not find that layout a likely one.