The default behavior of F3 in Nautilus is to open the Extra Pane, showing the same location as the first pane.
Is it possible to change this behavior or add something new which will open the extra pane always showing the same location, e.g. my home folder or the Downloads folder?
It currently isn't a feature of Nautilus to do this. Although it could be suggested as a feature for nautilus team to set up.
Nautilus used to have an option for it to start-up with an extra pane.
https://live.gnome.org/Nautilus is the place to visit.
They have a sections called "Ideas", that would be the place to suggest such a feature:
https://live.gnome.org/Nautilus/Ideas