I am a little confused by your description, because you say what you want to paint is covered by sheetrock. Typically, the sheetrock butts up against the framing of the window, and then you install molding over top of the sheetrock and the edge of the window frame, with a small reveal. After the molding is installed, you paint it and the window frame.

As to color, it is extremely common for those elements as well as the baseboards and crown molding to be white, but it is not unheard of for them to not be white.

To inject a little personal preference, all mine are white, but if I painted them a different color, I would want that to contrast with the wall color. That's why white is most common. Goes with every other color, so you don't have to repaint your trim if you decide to repaint your walls.

This is actually a fairly common question and I have seen it done both ways. It's really not called window trim though. A window may have casing around it (trim) or it may be recessed into the wall some and then you would have a drywall return there. Sometimes there is also a window sill on the bottom. I prefer to paint it the wall color because, well....it's actually part of the wall. It's easier of course to paint it white though because some may have difficulty making a straight line up against the vinyl window there. I prefer to use blue tape on the vinyl and then blade the tape down securely with a flexible putty knife and then paint.