It depends on the circumstances and the production. Specific shades or colours would sometimes be used because they would read better with particular types of film (This is especially common later, in television, because early television cameras had really odd colour properties). Sometimes sets would be painted in black and white, and other times set dressers would use off-the-shelf fabrics and paints.

It's important to note that even if you dress a set entirely in black and white, it won't look exactly the same on film anyway, and colour choices can help tune the contrast and highlights of a monochrome image. The norm has generally been to not make "grey" items specifically for film and television production, simply because it's more intuitive, but there are plenty of exceptions.

You may want to have a look at http://www.reddit.com/r/AskHistorians/comments/1tyawh/while_filming_bw_films_did_they_use_specific/ which has some comments (and videos) with examples.

Nope, they were in color. Sometimes, the colors were chosen less for actual aesthetics in person, and more for how they'd look in black-and-white. For example, here's the Addams' Family set.

I can think of one example when they did use black-and-white props, but it was for a special effect.

Remember the scene in The Wizard of Oz where Dorothy steps out of her black-and-white house into a technicolor world? It's actually a color shot of a black-and-white set. It's not perfect- you'll see it the next time you watch it- but it worked well enough.