Let's be totally honest with ourselves. We built our cottages to relax, kick off our shoes, NOT to supply ourselves with endless jobs for the rest of lives. It serves us best then to make the cottage and the finishes in the cottage as tough and resilient thus maintenance free as possible.
Thus using an appropriate finish for the doors makes a lot of sense. Consider the bathroom door - it will likely be, next to the front door, the most used door in the cottage. Lots of grimy hands, kicks, banging, and hot showers will be attacking that door. A tough coat of paint that is easily wiped down is the way to go.
Check with the professional in your local paint store or paint department in a big box store for you particular set of circumstances, but you can't go wrong with a gloss or semi-gloss latex paint formulated for bathroom or kitchen use.
The first paint on a new door is not an issue as the door will be primed, but for subsequent paint jobs make sure that you lightly sand the old coat of paint to make sure the new one sticks well. AND if you are going over an oil-based paint of some sort applying a primer paint may be in order.
With these few tips you will get a paint job that will last longer than you can remember the last time you painted...