Adding wallpapers in GNOME Shell

Is there any way I could set the [Wallpapers] tab to automatically find new pictures under /usr/share/backgrounds/ as well?

There's no such option. The [Wallpapers] tab shows the default choice of wallpapers (which may vary depending on distros/admins) available to all users. Distros/admins may choose to alter the content of /usr/share/backgrounds/ (along with the corresponding entries in the .xml file). I don't think it was meant to be customized by regular users.

is there a way I could change the configuration, so that, when clicking on the [Pictures] tab, it will look under /whatever/i/want/ instead of ~/Pictures/?

Yes, the [Pictures] tab is, to some extent, customizable. Pictures is one of the XDG user directories and is defined in ~/.config/user-dirs.dirs. Change the default value XDG_PICTURES_DIR="$HOME/Pictures" to XDG_PICTURES_DIR="/whatever/i/want", restart the shell and next time you click on the [Pictures] tab in the [Background] settings, it'll look under /whatever/i/want.
The downside is that some other applications that use XDG_PICTURES_DIR will also default to /whatever/i/want instead of ~/Pictures.

/usr/share/gnome-background-properties/desktop-backgrounds-default.xml holds the settings for the wallpapers; while there doesn't seem to be any command to rescan that XML file (one still can add further XML files into there, which keep an up-to-date index of some other directory).

however, one can define individual filenames:

gsettings set org.gnome.desktop.background picture-uri 'file:///whatever/i/want/background.jpg'

and one can access other parameters like that, as well:

gsettings set org.gnome.desktop.background picture-options 'centered'

when scripting wallpaper-changes, one has to set draw-background before and after:

gsettings set org.gnome.desktop.background draw-background false
gsettings set org.gnome.desktop.background picture-uri file:///whatever/i/want/background.jpg
gsettings set org.gnome.desktop.background draw-background true

there's also the gnome-tweak-tool which provides a GUI to these gconf settings.

@see Bug 682126 - Show the user's most recent photos.