So the further away the better....?
To a certain extent yes. 2.4Ghz is a very tiny wavelength. The object is well below the center of radiation. If it was parallel to the center of radiation I would say the more the better. On the bottom you just don't want the antenna inside to couple to the stand.
I am sure that we think of antenna differently. If you took the router apart the antenna could simply be a trace printed on the board with a matching network or it could be a metal loop. Either way it's very simple. The external antenna the attach to the screw on connectors (R-TNC or SMA for the little ones) usually just have a wire or spring inside. The ones that are fat at the bottom and skinny at top and claim some gain are 5/8 over 1/2 wave or some other combination of radiators that compresses the vertical pattern for horizontal gain (wow, just like the horn on a speaker, see why I love physics). Once you get into the directional yagi and log periodic style antenna do you see a complex series of elements.