to answer you question about free movement, the inner part of the caliper, the part that the pads fit into, should slide, slightly back and forth. As the piston pushes one pad against the disc, it pulls the other towards it.
It floats/slides on pins that connect the outer part of the caliper(the bit bolted to the hub) and the inner part (with the pads)
If you remove the pads, you'll see how this is meant to work, clean off any gunk and copper slip the pins up.
Also, clean up the pistons with brake cleaner and push them back a little, it should take a fair amount of pressure to push them back, but not stuck solid.
I suspect there could be other issues with your brakes.
Are the pads fitting the disc properly?
Do the whole face of the pad make contact with the disc? Is there a ring on the disc that looks like the pads aren't touching?
Are the faces of the pads evenly worn or are the wedge shaped or rippled?
Is the disc worn uneven, run you fingers down the face of each disc, is it all bumpy?
The above relate to worn discs, and the pads won't fit flush, so you won't get full coverage of the disc.
Does the car pull one way under braking?
Is the brake pedal squishy, how old is the brake fluid?