Can't identify this coil balance

I'd be happy to help, of course. Your pivot shoe looks like a match for our 15-111. This is specifically designed to work with the 1/2" thick Series 160 coil spring. We also sell the 15-150S single spring cover and 15-148 bumper, if you needed them, but you can always just reuse the originals.

Not necessarily. Some windows use one per side, others use two per side or more. You want to stick with whatever configuration you had originally just to be safe.