Because these territories were settled internationally, there were representatives from Southern states, but they found themselves in a minority. The western states remained fiercely loyal to the Union. A few people made it to the East to join in the fight (on both sides), but for the most part, the Far West was too far away to make a contribution with volunteers, and drafting its residents was impractical because of the distance.

Removal from the draft was part of what inspired Samuel Clemens to go to Nevada in 1861 (where he took his penname, Mark Twain in 1863). And it inspired others to flee to the West as well.

The Far West was able to contribute wealth to the Union cause. Nevada/Comstock gold and silver strikes in 1859 are often credited with "saving the Union." That's an exaggeration, but there is no question that the sudden influx of bullion into the economy helped offset deficit spending in the name of the nation's defense. The famous case of the repeated auctioning of the Gridley Flour Sack, which Twain mentions in Roughing It (1872), raised several hundred thousand dollars for the Sanitary Fund, the precursor to the American Red Cross: this is clear evidence of the eagerness of residents to contribute in some way to a conflict that they could only observe from a distance.

The Confederacy would have loved to capture the source of all that bullion, and this became the plot of the 1940 movie "Virginia City." But the core of that movie is pure fiction, and it was simply not possible for the Confederacy to reach that far as it fought, unsuccessfully, for its very survival.

Western newspapers are full of news of the war, but there was often a delay of often a week or more as word of battles eventually made it across the continent.

Don’t know about the other states, but I know a bit about Oregon from that time.
Oregon was only a for a couple years before the Civil War broke out , so a lot of the territory was still managed by the federal government, & patrolled by the US army. When war broke out all the federal troops were called back, leaving big chunks of the state essentially lawless. Troops were called in from California & Washington, and a cavalry regiment was organized in Oregon.
On the West side of the state things were pretty quiet, only fighting actually happened after the war was over. When news of Lincoln’s death reached the state, Philip Henry Mulkey walked the streets of Eugene & repeatedly yelled “Hurrah for Jeff Davis, and damn the man that won't!” He was quickly arrested. A group of Union sympathizers wanted to hang him & attacked the jail. An armed group of Mulkey’s supports from Long Tom arrived & squared off with the mob & violence was about to break out. But the local infantry division slipped Mulkey out of town & held him in Fort Vancouver until things could cool off.
The eastern half of the state saw much more activity. There was a bit of a gold rush going on out along the Oregon Idaho border and miners were pouring into the region. With no army to patrol the area & keep order, all the new miners were causing problems with the Native tribes, trespassing into reservations, fucking up the rivers, also fighting with each other, and generally causing problems. The tribes, for their part, took to raiding mining camps & stealing any supplies & horses they could get their hands on. Cavalry patrols were set up on the main routes in & out of the goldfields, and around the reservations.
Many of the Cavalry groups did their job well, but others saw it as an opportunity to make a quick buck. The Olney Detachment patrolled a stretch of the Columbia gorge that connects the West & East halves of the state. They basically turned the assignment into a protection racket, only defending groups that would pay them. They also scammed the army out of equipment & gear that they would then sell off in different towns & pocket the cash. To the army’s credit, they only got away with it for 3 or 4 months before the detachment was disbanded.
Elsewhere in the state, tensions between the tribes the settlers got worse. The army was stretched thin and couldn’t intervene to keep the peace like they had before. Near the end of the Civil War, open war broke out with the tribes. Cavalry raids would be sent out to sweep up “rogue Indians” but the Indians could move faster and disappear into the canyons & hills. The Indians would raid settlements, farms, mines, any target of opportunity, then disappear back into the wilderness. Fighting went off & on for 15 years.

Not much did happen west of Texas. By far the largest campaign was the New Mexico campaign of early 1862.

The southern part of that territory was more closely associated with Texas and felt that the government in Santa Fe wasn't paying proper attention. They formed a small volunteer force that was suplimented with several Texas companies. All told General Sibley had about 2,500 men.

Opposing this was a force of US regulars, several volunteer companies from Colorado, and the New Mexico militia. All told the Union had about 5,100 split between Colonel Canby at Fort Craig SW of Santa Fe, and Colonel Paul at Fort Union east of Santa Fe.

The Confederate plan was to march up the east side of the Rockies, capture Santa Fe then Denver and then Laramie. This would cut all the major trails between the East and West coasts. It would also capture the gold and silver fields of Colorado. Follow up campaigns were to include raids into Nevada and drives to capture Southern California and Northern Mexico. Sibley was also planning on doing all this with a minimal supply train, planning on using captured Union forts and depots. It makes one wonder if Sibley had any idea how big the West was, and how hard it would be to "live off the land" in arid environments.

In any case Santa Fe had to come first. Initially Sibley advanced well. He captured Tucson and advanced toward the larger Union force at Fort Craig. He needed to reduce it before moving on. So Sibley moved toward the Fort. Canby was game and moved to block Sibley. In a couple of sharp engagements Sibley beat the Feds and drove Canby back to the Fort. By this point Sibley was running low on supplies so he decided to make for Santa Fe and cut off Canby's supply line.

Canby took all his cavalry and started raiding in front of Sibley's advance. The idea was to slow Sibley down and strip the country of anything the Confederates could use. It worked. Raiders weren't sufficient to stop an advance though, and Sibley took Albuquerque and Santa Fe. By this time Paul in Fort Union had been reenforced by more companies from Colorado by Colonel Slough. Paul had been willing to wait in Fort Union for the Confederates. Slough was a more aggressive commander and took control of the fort. He quickly organized his forces and advanced to meet the oncoming Confederates.

They ran into each other at Glorietta Pass. Slough split his forces and attempted to surround the Confederates. The Confederates however weren't where they were suspected so instead of hitting them with a pincer Slough was forced to face them with only a partial force. In a couple days fighting was forced to retreat back to Fort Union. The detached force however ran across the Confederate baggage train. They took or burned 80 wagons and captured or drove off 500 odd horses and mules. So even though they won the battle, they were completely unable to advance. They had to fall back to Santa Fe, and shortly after retreated to San Antonio. Canby skirmished with them several times to keep them moving back.

In all the Confederates never had the numbers to properly hold the territory. By the time of Glorietta the main force only had 1,100 men. Casualties and the need to garrison captured towns had cut Sibley's force to less than half. Paul and Slough each had similar sized forces in front of him, and Canby was chasing around the territory with a couple hundred horsemen harassing him, and had a larger force of infantry ready to hit him from behind.

After the campaign an additional 1,000 men advanced from Southern California. With the additional men Canby drove off the remaining partisans, captured the capital of the Confederate territory of Arizona and advanced to El Paso. That was too far removed from the rest of Texas to make a serious attempt to invade Texas from the west. So with the capture of El Paso the war in the Far West largely came to a close. Union forces in the area spent most of the war fighting Native Americans and the occasional partisan band.