Unlike rim brakes, caliper brake is moving brake pad on one side only - the opposite side is stationary (at least the one I have is like that). What I wonder is if piston/pad is pushing/bending the rotor towards the opposite pad or something else is flexing/giving in so brake pads can clench onto the rotor?

This pretty much sums it up..

The outer pad moves toward the rotor when the caliper-actuating arm is pulled by the brake wire. The inner pad can be adjusted toward or away from the rotor with a pad-adjusting knob, but it is fixed during braking. The moving pad flexes the rotor toward the fixed pad when the brake is operated. The moving outer pad also uses an adjusting knob to position the pad relative to the rotor. The dial uses an indent system, with 16 per revolution. One complete revolution moves the pad approximately 1mm.

Less-expensive disk brakes are like that. Better brakes (such as Avid BB-7) move both pads.

bb7 is single piston as with most mechanical disc brakes. there was a mechanical with 2 pistons but i think it is off the market now. to get 2 pistons you need to go hydro. hayes did have a single piston hydro but that is obsolete.

I don't know anything about bike disc brakes/calipers, but most cars have one or two caliper pistons on a single side, and then the whole caliper slides on pins, thus allowing the brake pads on both sides of the rotor to move toward or away from the brake disc. Only cars with high-end braking systems (Brembo, etc) have pistons on both sides of a fixed caliper.

Are there any bike brake calipers that slide on pins in a similar fashion?

yea the ones on walmart bikes