Addendum: I just re-enabled the terminal, right-clicked on it, opened the preferences window and disabled 'Override Geany keybindings'. This seems to prevent the terminal grabbing the function key presses. So, give that a try if you don’t want to use MultiTerm (but as I said, I think MultiTerm is better anyway).

Johannes, on my system the function keys stop working even when the terminal doesn’t have focus (unless I override the keybindings as above).
James