One question I have is about UI representation of toggle button to turn on/off this behavior. In this patch I implemented checkbox with text, but there is also possibility of less cluttered icon toggle button to the right of the normals size slider.

"Constant Screen Size" seems reasonable. Would not use "length" since it's already named "size" and that's the term we use for other viewport preferences.

I think an icon is preferable, but maybe the UI team can help find the best icon to use or add a new one.

Regarding the icon
I created my version of an icon to be used in this patch, unfortunately I didn't manged to include it as whenever I saved the blender_icons.svg file my Inkscape version was changing a lot of stuff all over the place in the SVG/XML text file and I think this was affecting the blender_icons_update.py script as it wasn't adding a new *.dat file for the icon but also was changing some other random ones.

First things first - UI/UX
@Julian Eisel (Severin), can somebody from the UI team can give "go/no go" for this UI/UX approach presented in this patch (video showing it is in description of this diff)? If this solution will have the blessing from the UI team then I would propose to submit the icon in the separate diff (before this one). Of course as discussed with @Julian Eisel (Severin) this new icon would need to be accepted by @Andrzej Ambroz (jendrzych), but we can talk about the details in private. My current version looks like this (in the center):

Regarding the design: Personally I too prefer having it as an icon, simply to make it take less space. Although it's certainly useful, it's not a vital option, so having it nicely compact makes sense.

@Piotr Makal (pmakal) yeah, unfortunately Inkscape always does change a lot of stuff in the SVG. This stuff can be manually removed from the patch (e.g. by only adding relevant bits to the commit using git add -p), but it's a hassle. And it's not always clear what's related and what not... (usually it's harmless though, it's just a bit of noise in the patch.)
Also unfortunately, there seem to be version differences in Inkscape that generate different .dat files. Just use git restore path/to/a/file.dat for the unrelated ones.

Hey - this button cries for more generic and universal "fixed size" icon, that can be easily reused enywhere else in the GUI.

(just ignore the right one - it's way to similar to an "X"=delete)