• I've changed grid colors but they're not showing. What should I do to change that?
• How can I fixate that grey-white checkerboard background so that it's always the same 1 pixel height and width?
  If I move and scale the canvas, checkerboard changes.

https://i.imgur.com/UhfHWti.jpg

Grid you're seeing is the pixel grid, not the custom grid you set up. See that your grid has 20 pixels in x and y dimensions, which means the lines are on the edges of your canvas. If you want to use this kind of grid for pixels, you need to set it up to have 1 pixel in every dimension.

But it would be just easier to go to Configure Krita -> Display -> Grid Settings -> Pixel Grid and choose a different color there.

Transparency checkers are just for transparency, not to support pixel art, at least for now. Please try Configure Krita -> Display -> Misc -> Move checkers when scrolling (check it) and see if it helps. If not, I'm afraid I don't know of a setting that would help.