I wouldn't paint it at all. I like it.

The door looks quite nice and I wouldn't screw up its appearance with paint or a chalkboard. Maybe there's a better location for a chalkbd/bulletin bd like a mud room or laundry room.

Velcro.

You can use a pealable command hook on either the inside and the hemp rope would be coming over the top or placed on the outside for the look below with an old tray (or any sized/shaped board and that also could be framed) chalk painted and drilled for the rope. Not suited for lots of writing though - but cute and door is still appreciated. If it were me, instead of covering too much of your lovely door if wanting something large, maybe the nearby wall would suit.

Liquid Nails would work if you do not plan on taking it off anytime soon!