Gentlemen, I tried your advice. I did not have a helper near me, but I did pump the brake pedal 3 or 4 times with caliper removed. Then I opened the brake reservoir cap and used channel locks to compress the piston back in. Piston did move in and out easily without any obstructions. There was no any signs of dirt or rust on the piston at all. It was nice and smooth surface. Anyway, I lubricated it with the PB blaster, and installed everything back. I also cleaned and lubricated sliders. Now here is the funny part: obviously piston moves in and out easily, but when I installed everything back and pumped the brake pedal several times there was no pressure on the brake pads of the mentioned caliper at all, and caliper can still be moved left and right. Very weird. I did not have anyone around me to help me with brake bleeding this time, so that will be my next attempt, but the way everything is going, I already have a feeling that line bleeding will not help. Is front left line diagonally connected with rear right or? I also wonder if this problem had anything to do with my shifting problems since clutch line and brake lines share the same fluid.