How can I monitor other windows non-intrusively (enable picture-in-picture feature) on GNOME 3?

Solution 1:

The GNOME Shell extension Window Corner Preview is exactly what you were looking for! Follow these instructions in order to install it.

With Window Corner Preview simply select a window or workspace you wish to preview:

And you will have the window in the top-right-hand corner (although if you left-click on the window preview you can make it go to the opposite corner, right-clicking the window preview will cause it to go to the next corner along in clockwise motion).

If you wish to simply see what is behind the window preview just cursor over it and it will go mostly transparent:

To hide the window preview just use the On/Off slider you saw in the first screenshot which can be used once you have made a window or workspace selection from the list.