If one do not want to use a paint stripper to remove the remaining paint, is it recommend to paint a new water-based wood paint on the door?
If the existing paint is latex there's no problem. If it's oil based it may take a little more prep. For either, a light sanding to knock off any surface shine and loose paint, then a light wash to remove sanding dust, would help the new paint adhere better.
While it would be nice to know if it has lead based paint now, it doesn't change the general prep process. It's only a major concern if you plan to strip the door back to bare wood, which generally isn't necessary.