Ubuntu – How to monitor other windows non-intrusively (enable picture-in-picture feature) on GNOME 3

There are some GNOME shell extensions which may help you achieve this goal.

1. Window List
   It displays a list of open windows at the bottom of the screen. You'll be able to minimise and restore windows by clicking on the window names.
2. Window List Mod
   It's a modification of the Window List extension. It displays a list of open windows at the top of the screen and offers a few more customisations.

To an extent, you have to adapt to the working environment you choose to work with.

1) a grid of workspaces (3 columns, 2 rows to be precise); You already found the Workspace Grid shell extension

2) the application windows remaining in the workspace I move them to; Using Gnome Tweaks, which is not installed by default in Ubuntu, you can set workspaces to "Static". With dynamic workspaces, workspaces automatically are removed when they are empty, so indeed an application you placed on workspace 3 may end up on workspace 2 or 1 later in the session.

3)miniatures of the workspaces content in the switch modal; Gnome-shell provides miniatures of the workspaces in the overview when you press Super. By default, however, you have to hover the right side in the overview to make the miniatures visible. You can make sure the workspaces are always expanded with the extension "Always zoom workspaces"

4) workspace switcher showing the overview of all workspaces. I do not follow here: in the overview, all workspaces are shown. You may also install a "Workspace switcher" extension that provides a clickable icon on the panel that allows you to switch workspaces from a pull down menu.