How to get precise borders and sizes in Android KitKat's Webview(Android Chrome) on Nexus 7?
This is not a bug.
I tend to think of 1 CSS Pixel to being equivalent to 1 Dip in a Native Android App.
In the web you aren't expected to cater to screen pixels, same as you aren't expected to use screen pixels in native applications (regardless of the platform).
How should I set the border, so it looks the same every time and it's hairline thin
The whole point is that it should look the same regardless of screen size. You may have more luck using an image and applying it as a border image: http://www.w3schools.com/cssref/css3_pr_border-image.asp
Or there are some hacks which exist for retina screens that may work for the Nexus 7 resolution: http://bradbirdsall.com/mobile-web-in-high-resolution