62/180 is a very good highway with no issues. We've stayed at Brantley Lake State Park for touring Carlsbad.
The elevators are now in service. However, on the web site at the top of the page is an Alert so notice that they've had rain which caused some flooding and more rain is in store so check first as you get closer to El Paso.
https://www.nps.gov/cave/index.htmYou mentioned you're coming from Arizona and there's another gorgeous cave - Kartchner Caverns State Park at Benson, AZ just off I-10 that you might want to check out. It also has a nice campground with electric. (Volunteered there)
http://azstateparks.com/Parks/KACA/Caves are nice to explore in hot weather. :) Enjoy!
