What size are your background tiles? If they're not power-of-two square then that's likely the problem for the blurriness and misalignment. It's likely your video card.
Most video cards are optimized to handle textures that are power-of-two squared, and any other size causes these irregularities. This means that for a seamless, repeating texture you will need to use Tiled Backgrounds that are 8x8, 16x16, 32x32, 64x64, 128x128, etc. So if your background tiles are the same 25x25 as your player sprite, then you will run into problems.
This isn't a Construct issue, it's a hardware issue. Due to how bits work in computational terms, it's industry standard to conform to these because computers can crunch those numbers more efficiently. Some graphics cards though are designed to handle non-standard texture sizes like 25x25, but apparently that's not the case with yours.
If you redesign your tiles and sprites to conform to power-of-two sizes then you will probably see the results you are looking for. Yeah, it might be odd doing math for them at first because you can't do it in your head as easily as with factors of 5, but you'll get used to it eventually.
As for the grid shown in the layout editor, you can change the grid size to whatever you want
Apart from that, I can't quite tell from your post if your player sprite is moving how you intended... something about only appearing to move at the edge of the screen? I don't quite understand what you mean. Perhaps you could post a .cap showing what's going wrong, and explain in greater detail what the effect is that you're looking for.