Is there a hotkey to toggle between the two panes only in Nautilus (with Extra Pane/F3 mode)?

F6 switches between the two panes. You don't need to use Tab and Shift + Tab. This works on 10.04.

If you are using Status Bar the control will be to the first element on it. But if you remove the status bar by choosing "Status Bar" in the view menu, the control will move to the right side file browser's column when pressing the Tab key.

Shift + TAB will take the control to the left side file browser's column when in the right side.