I'm interested to know whether El Paso was actually taken from New Mexico and given to Texas to pacify slave states after reading the Wikipedia article History of El Paso, Texas - Texas independence, which says that:
The area was never considered part of Texas until 1848... Given the blurry reclamations of the Texas Republic that wanted a chunk of the Santa Fe trade, the Treaty of Guadalupe Hidalgo effectively made the settlements on the north bank of the river a formal American settlement, separate from Old El Paso de Norte on the Mexican side.
Most importantly, the article makes the following claim with no clear citation:
The present Texas-New Mexico boundary placing El Paso on the Texas side was drawn in the Compromise of 1850.
If these claims are all true, would that mean that the reason why we have El Paso, Texas and not El Paso, New Mexico is because of a broader effort to pacify slave states and forestall the Civil War?
To say it was 'taken' from New Mexico is... not entirely correct. The Republic of Texas had always considered its borders to be the Rio Grande in the south, and the Arkansas River in the north. While it was unable to project its power that far, and Mexico only recognized them within the confines of the Red River (north) and the Nueces River (south). Admittedly, Mexico also wasn't really able to project power and authority in the disputed territory, and the locals were largely left to fend for themselves.
These competing claims led to incidents like the Sante Fe Expedition in 1841. Mirabeau Lamar, who was nothing short of an utter failure of a president, sent out a large merchant caravan with a small armed escort to arrive in Sante Fe and try to dominate the local economy and even negotiate the official transfer of the territory to Texan authority. The Mexican governor was not amused, captured the lot, and after debating the wholesale execution of the group including the civilians, the present council elected instead to imprison them in Veracruz. The US had to negotiate for their release, including the famous Henry Clay, who would later become instrumental in the Compromise of 1850.
Speaking of the Compromise of 1850, yes, El Paso technically belongs to Texas because of it. No, it wasn't "taken" from New Mexico. Yes, it was originally owned by Mexico, who actually organized it as part of Chihuahua, rather than Nuevo Mexico. When the US annexed all the way to California, it also went in to annex the territorial claims of its newly acquired Texas (which was annexed only a couple years before the Mexican-American War). That is to say, it went to the Rio Grande, instead of stopping at the Nueces and saying "good enough". Post-annexation, the US had some reorganizing of territories to do.
The Compromise of 1850 included Texas in its reorganization efforts. A handful of proposals were made on where, exactly, the Texas border should rest. Henry Clay, mentioned earlier, and famous as the most unelectable American of the 19th century somehow, was one of the people attending - but his plan was not used. Instead, one of the more restrictive plans was put into place, with Texas retaining territories to the Rio Grande, but abandoning claims to New Mexico (which also includes parts of present day Oklahoma and Colorado) in exchange for the US assuming the rest of the debt Texas held as an independent republic. This Pearce Plan, named for its architect James Pearce, is the heart of the whole thing.
You could, technically, say it was part of the efforts to pacify slave states, but on the whole here, in the specific context of Texas, it dates to an older Texan-Mexican issue, and the proudest and slaveholderiest of the Texans didn't really get what they were hoping for when territorial allotment came around.