Do not caulk each board to each other- the overlap. But do caulk the but joints and against any trim, windows and doors. You can paint but if a lot of the original "paint" comes off you can use a solid colored stain. Stain has advantages but only if it is applied to raw unpainted wood, over paint a solid stain is no better than the underlying paint.
If you decide to paint definitely use a standard oil base wood primer. This will soak into the wood. Even better is thin the primer a bit (so it soaks into the wood) and apply 2 coats then paint.
Go ahead and wash then scrape and see what comes off then proceed accordingly.