I live in the upper sonoran desert. Here Russian sage has already bloomed, and my shrub has lost its first flowers completely because it's in a hot west facing location. If you cut off the top of each branch to just where the first flowering is now faded, will it encourage another bloom or is it not advised because it will bloom lower down the branch as summer progresses anyway?

Yes, if you deadhead the Russian Sage, it will bloom again.

And if you don't deadhead, then the plant is not going to bloom again? Around here, I see landscapers continually cutting back the mature Russian sage shrubs 2 - 3 times throughout the summer. If they didn't do this the plant would only bloom once?

Aound here they only bloom once because no one bothers to do anything with them. It could be self-defence on the part of the gardeners there - My biggest weed nuisance (other than cheat grass) is Russian Sage. Your gardeners don't feel like pulling all those baby Russian Sages next spring.

The RS should be cut back after they go dormant. Otherwise, all that dead stuff is still there amongst the new growth. Around here, they are really a nuisance plant. Idaho has outlawed them and put a bounty on the ones that are wild in the desert.