Truly good questions the kids have been asking and I'm looking forward to many of their questions arising from "Wink of an Eye" and others.
1) There is no suggestion in TOS that the Bridge does have a bathroom / toilet as the wall panels left and right of the main viewscreen do not feature handles or anything that could suggest such a thing.
Either Bridge officers "do their business" prior to their work shift (with the help of 23rd Century medicine?) or can ask one of the other Bridge crew members (from secondary monitor stations) to take over for a few minutes, IMHO.
2) The way I recall it there was no button pushing scheme on the Bridge, except for Kirk's intercom button on the command chair.
However, for TMP / ST I they went for a scheme where the actors had to familiarize themselves with the console layout and push the right buttons to provide more realism.