In NV, fingerprinting is done by the county sheriff when you apply, it's part of the application fee. The only States that allow you to mail in your fingerprints that I know about are CT, FL, UT and VA. Some of those States (FL and VA) require that the prints be taken by a US law enforcement agency in which case I include the receipt they provide stating what agency took the prints.
Many instructors provide the application packet as part of the class materials along with instructions on how and where to apply to the county they are based in.
One day for the 8 hour initial permit class and count on a day at the LVMPD CCW Detail. You can call the CCW Detail and they will mail you an application package along with their list of approved instructors. Washoe and Churchill Counties should do the same thing. However if you're not a Nevada resident, I do recommend you apply through Washoe or Churchill counties since they tend to be quicker and more efficient than Clark County. You can legally take your training from any county approved instructor so it is feasible for you to get your instruction here an apply in Washoe or Churchill counties. Per the NVSCA it is the responsibility of the sheriff (or their designee) you apply to contact the county where you received your instruction to verify the instructor's status if you were not instructed in their county. However, Washoe and Churchill counties now require 4x4s and snow chains or tires to get to after flying into the closet airport.
You will also need your green card if you're not a US citizen to apply through any county.