(1) Has the question about a bathroom on the bridge ever been answered?
Never officially, but the Franz Joseph blueprints of the bridge had it ringed with a corridor between the outer shell and the bridge, accessable via the secondary door seen in The Animated Series
. This corridor did include toilet facilities.
The Next Generaton
went one further by actually including a door labelled 'head' at the back of the bridge, just next to the doors to the conference room.
(2) After seeing several close ups of Spock and Uhura pushing and manipulating buttons at their respective stations, my son wants to know how they know what buttons to push without any letters, labeling, etc. Did the actors actually have a system/routine that they followed, or did they just random push buttons in the scenes?
I do remember hearing that actor George Takei worked out his own method for operating his console as Sulu. Perhaps the other actors did the same? Spock's station was by far the most 'hero' of them all (ie, it had the most props to play with, like that sensor thing he was always looking into) so I imagine Nimoy at least figured out in his own mind what everything did and tried to keep it consistent.