Ok, this is now in the main section.
If the air pump can't be turned by hand it is apparently is seized so there's no reason to put a belt on it. It injects air into the exhaust to help it burn when passing through the catalytic converter, so it isn't absolutely necessary. But, the truck wouldn't pass emissions testing.
As for adjusting the belts, I like to see ~1/2" of flex when pushing firmly with my finger. But that is not at all scientific. And I'd rather have a belt a bit loose than a bit tight since too tight can hurt the bearings.
Also, when you have two belts on the same device, like your alternator, they should be bought as a set or pair. Belts aren't held to a very close tolerance and buying two run of the mill belts rather than a pair will frequently have one of them tight and the other loose. But it is hard to find paired belts.
Gary, AKA "Gary fellow":
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Dad's: '81 F150 Ranger XLT 4x4: Down for restomod: Full-roller "stroked 351M" w/Trick Flow heads & intake, EEC-V SEFI/E4OD/3.50 gears w/Kevlar clutches
Blue: 2015 F150 Platinum 4x4 SuperCrew wearing Blue Jeans & sporting a 3.5L EB & Max Tow
Big Blue: 1985 F250HD 4x4: 460/ZF5/3.55's, D60 w/Ox locker & 10.25 Sterling/Trutrac, Blue Top & Borgeson, & EEC-V MAF/SEFI