Folding bead makes it easier to carry a spare tire with you for touring. Otherwise it dosn't matter.

Your rims WILL take a 1.25" tire no problem.
I've had rims marked 1.75 that were narrower than one marked 1.50.
That's just a "nominal" tire width.
One of the rims on my grocery getter is an Arraya VP-20 marked as 1.50. The 1.25" tire looks like it "belongs".