Can somebody check the attached video, please? Is this noise (the pedal comes slowly back, and sound like pumping a ball with air) normal if I'm releasing the brake pedal? I've already replaced the brake discs&pads and the brake fluid, but the problem still exists.

 

Are you refering to the rattle/thunk when you release pressure? That is likely just something loose in the pedal assembly, check the bolts on the pedal arm, especially the one connecting the arm to the master cylinder.
As for the squidgy noise when applying the brakes, do they feel nice and solid or do they feel like stepping on a sponge? With the car off the pedal should become very stiff when you pump it a few times. If it doesn't you may have some air in the system and you need to bleed the brakes again.

 

I think the pedal assembly is ok. My problem is the squidgy noise when I'm releasing the pedal. It comes "slowly" back and it sounds really weird. If the car is off the pedal is very stiff after a few pump.

Maybe a better video about the problem:

 

Sounds normal to me. The brakes are hydraulic.

 

Yea, sounds normal to me also.

 

Is the engine running when you made these videos? it's normal for the pedal to get stiff after a few pumps. I don't understand all the details but the brakes are aided by the brake booster which uses vacuum to amplify the force from your foot to the brake calipers. If the engine isn't running though, there's no vacuum so the brakes will behave differently.

these videos seem normal to me, but if you're concerned about it, I'd have a mechanic check it out sooner rather than later, things rarely go well when the brakes fail.