I think it's more of a faux pas not to use the same white for all the trim in a room, if you're painting all of the trim white. If a room is small or has several doors and windows, consider painting the trim (and doors, if desired) the same as the wall color so the walls aren't as "interrupted" by the trim.

As for the paint finish, I typically use semi-gloss for trim since it's more durable (moisture resistant and scrubbable), and trim tends to take more of a beating than walls do. If semi-gloss is too shiny for your taste, you can do a 50-50 mix with eggshell or satin finish paint in the same color to cut down the sheen.

Yes, a slightly lower gloss can be better in average sized rooms. And of course there are seventy-leven zillion shades of white...the trick is to find one that works with all your wall colors...and, yes, all your trim should match. The exception is if you want to paint the trim in a room to match the walls. This can work with a rich dark color, like a deep blue or a rich clay in a den or library or powder room, to minimize the distraction of a lot of trim. But I think that works best in a room that is off to itself, rather than one that is visible from several other rooms that do have white or cream trim. And it's not to everyone's taste.