When I set resolution at 800x600 DirectX 11 in fullscreen, the picture is stretched. How can I fix that to get a proper aspect ratio with black borders on sides?
Also, for some reason compatibility settings are grayed out in Properties.

My guess is that your monitor and GPU software are both not set to preserve aspect ratio on scaling, so it fills the widescreen display, stretching it.

If you monitor supports scaling then go into your monitor on screen display settings and tell it to display as 4:3.
More preferable, however, would be to set your GPU software to perform scaling on the GPU and to preserve the aspect ratio.

Why though? Even my GPD Win (Atom-based SoC) can run Blood Fresh Supply at its native res (1280x720) at 100+ fps.

If you want to run at lower resolution for "authenticity" you're better off using either DosBox or NBlood with the classic renderer.