This clickpad requires some manual Xorg configuration in order to get right and middle clicking to work properly.
Create a new file called /usr/share/X11/xorg.conf.d/52-synaptics-np900x3b.conf.
Inside paste the following:
Identifier "np900x3b clickpad"
# Enable the clickpad and set click actions
# Single click = left button
# Two-finger click = right button
# Three-finger click = middle button
Option "ClickPad" "1"
Option "ClickFinger1" "1"
Option "ClickFinger2" "3"
Option "ClickFinger3" "2"
Then log out and log in. You should now be able to right click by using two fingers, and middle click with three fingers. I have not experienced the one-finger double-click problem that you describe. If this configuration doesn't fix it, you could also add the option
Option "SHMConfig" "True"
Then you can try debugging with
synclient -m 1 which will give you feedback on what the touchpad driver is seeing.
More information about this fix and other issues are available at the Series 9 Community Wiki page