I am doing guitar setup for the first time and I think I got little bit lost. I lowered action on my schecter but I think the sound got worse - totally lost smoothness. Do I need to lower pickup height? There are 6 screws on the pickup cover on one side of each pickup - if I lower these would the whole piskup go down or only that side?
NO! Dont touch those, those are the polepieces, the actual magnets for the pickups and you shouldnt need to adjust those right now.

There should be screws on the side on pickup rings or in the cavity if they are direct mount, just tighten those to lower it
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thanks for reply. I get it now.
Do you know about its bridge hight adjustment. It wasn't easy to lower it and it is very hard to put it up. I was doing it with my fingers but it seems that I can use a screwdriver. It's hard for me to describe it but it's not like on LesPAuls, here each side of bridge has actually a screw head that moves as I change bridge height. Do you know if it;s OK to use screwdriver?
You can adjust the height pole piece. It's fine.

And in a humbucker, pole pieces aren't the "actual magnet". There's a magnet inside the pickup.

Here's a guide on adjusting pickup height:

And of course it's hard to raise it, you've got the full tension of the strings on there.
Detune your strings to relieve the tension, and try again. And yes, you can use a screwdriver, but loosen it the strings first so you don't chip the metal.