Consistency is best, but there can be a certain consistency in non-uniformity. Are the stained windows within one room? Are the french doors? One option that you have that also has historical antecedents is a stained door with white trim, windows/ trim, and baseboards, and you could do this throughout the house for a certain amount of consistency. If the windows with the stained trim are all in one room, perhaps that entire room could have stained woodwork. Note that doing stain grade doors throughout the entire house can get relatively pricey. The key is to have a rather consistent plan or hierarchy and following it.

The following steps are for stained trim/beadboard. Painted trim/beadboard is covered in the last paragraph. In your case, installing the door trim first is step one. To get the base molding to match the door trim, it will need installing second, unless you cheat by planing the base molding thinner to allow it to match the door molding when installed over the beadboard. That will make the base molding thinner and it will be different looking in width from the door molding---but the surfaces will match. That generally is not acceptable to most people. That poses a double problem, however. That problem is how to get the edges of the beadboard to match the edges/bottom of the door and base molding. One way to eliminate this problem is to cut rabbets in the door and base molding. A rabbet is a groove cut into the side of a piece of wood and forms a notch the whole length(or as much as needed for the beadboard on the door molding sides) of the wood. That allows you to fit the edges/bottom of the beadboard in the rabbets and elininates any gaps. The only gaps will then be in the gaps created by the pattern of the beadboard. A colored caulk can be used to fill those if the moldings/beadboard are stained. If you are painting both trim and beadboard, it is much more simple. Install both moldings first. Then install the beadboard. Then caulk the gaps with a paintable caulk, prime, and paint.

I’m looking to paint our walls Sherman Williams Alabaster or Pure White? We are still deciding. What color white should I paint the baseboards and doors? What about the ceiling? I’m worried about choosing a white that clashes.

Are you asking about trim for the dining room or throughout the house? If you are specifically asking about the dining room, do you think you'll ever add wainscoting? If so, you'll need to pick trim thick enough to blend with the wainscoting/chair rail. We just added wainscoting to our dinign room and had to replace all of the door and window trim for this reason.