I bought a 1911 two storey house that has a slightly newer addition - I don't have a whole lot to spend at this time but the trim in the original part of the house is original dark stained wood while the trim in the addition (which includes the kitchen and dining room downstairs and the bathroom and a bedroom upstairs) is painted and is essentially just wood boards. I was thinking of adding a small piece of trim to the boards to make it look more like the rest of the trim, but should I paint it all white or paint it brown to match the trim in the main house? Or will it just look stupid and I should work on replacing all the trim in the addition?
This is really based on your personal taste. To make the addition look less like an addition match it with the other trim. Without pics I will venture a guess that to just add something on top of the present board will look like a patch.
I believe your orginal wood trim was not stained but has 3-4 coats of orange shellac applied. Over time air and sunlight reacts with the shellac and turns it dark brown with a touch of red. Orange shellac was the finish of choice during this time frame and was made by adding 3-5 lbs of powdered orange shellac to a gallon of alcohol.
Orange shellac and white shellac (clear) is still available but it is almost impossible to match with 100 year old shellac. Old shellac can be removed with a rag soaked in alcohol.
If it was me, I would paint your additional top trim piece.
If you try to remove your existing baseboards, you will find the baseboard extends below the hardwood/pine flooring. A very tough job, indeed.