Hi, Catherine. I don't have a problem with stained woodwork in the bathroom. Just install it, stain and a couple of coats of poly and you should have no trouble. We have oak accessories in our bathroom and the light switch plate, tissue holder and towel bars have held up well. But the water glass/toothbrush holder is kind of a disaster because it is always wet. But this is no big thing because it is cheap and easy to replace. Your woodwork should not be subject to the same treatment and not cause a problem.
As for the kitchen, there are plenty of kitchens with wood floors. Jay J knows more about wood floors than I do and you should get his input on this.
But I just can not tell you to put a wood floor in the bathroom. With 4 kids and 2 grandkids, I know that you just can not help but have water on your bathroom floor and no wood will take that kind of abuse. I know that there are wood boats, but even those were subject to a great deal of maintenance, like annual painting or varnishing. I don't believe that you would want to do this and the next people who live in your house would certainly not want to either. Catherine, I would sure suggest that you find something else (marble, stone, slate, terrazzo) that will stand up to water. Find the floor covering store in your town that caters to the expensive alternates and visit them to consider what options are best.