I tried to use grid and other ways, but nothing seems to resolve.

https://ibb.co/5Kq4yJv

Inkscape is just too expensive to use, as you can see, things simply do not work and require a lot of hardwork to make simple sharp lines and sharp rectangles.

vaidas.boqsc

Inkscape is just too expensive to use, as you can see, things simply do not work and require a lot of hardwork to make simple sharp lines and sharp rectangles. 

So, are you looking for a solution, or just here to complain?

I see in the screenshot you are working in mm... have you considered using pixels as units?.

A simple Google search "inkscape snap to pixels" has a solution in the first result.

The manual also has a solution... have you explored the manual?

http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Extensions-ModifyPath.html#Extensions-Pixelsnap

Adjusts paths and rectangles to the pixel grid so that when exported as a bitmap, the lines are sharp. Filled objects are adjusted so that their edges are aligned with pixel boundaries. Paths are first adjusted to have an integer pixel width, and then adjusted so the stroke edges align with the boundaries.

I'm here to complain and find a solution. An extension is not a solution it's still a hard manual work unlike something like grid system that guides you through,
but it seems to not produce what's needed no matter what settings I throw at it. 

I've tested and tried all the things you mentioned before, I wonder what's the root cause of all this.

From this topic: https://inkscape.org/forums/questions/get-good-answers-here/

1. Describe the specific steps you took, the result you expected, and the result that actually happened.
2. Describe your OS (your computer's Operating System: Mac/Windows/Linux/etc. & version).
3. Describe the version of Inkscape and the installer you used (the file you downloaded from the website: .msi/.exe/.zip/other ). 
4. Describe any special hardware being used, like tablet/stylus or multiple monitors.

• Attach* an example SVG file that has the issue.

Not completely sure about your issue but your grid doesn't follow even pixels either on the x direction -spacing unit is 1.1 px- in your screenshot. 

Thus vertical lines would appear "px perfect" if the nodes are placed on every 10th grid line.

Blurriness is caused by anti-aliasing. Even if those rectangles aren't rendered blurry aligned perfectly on a horizontal/vertical grid, if you rotate them they will be "blurred" slightly. 

Solution?

Switch off anti-aliasing if you are good with jagged edges.

If not, oversample. Make sure your lines are at least 6 px wide at 100% zooming. Then it hardly even matters if anti-aliasing is on or off -by the assumption you have the right screen resolution.