In my opinion, that room is quite adequate for either of those instruments -- and I don't think that adding a door would be a bad idea at all. If I may suggest it, I would place the piano so that, if the door is open, the player can at least glance through the door -- but that's a personal preference.
I would also seriously consider an area rug, assuming that there is a hard floor -- wood, tile, what have you. I might also consider placing a tapestry or other fairly heavy hanging on one of the walls.
I might also add that you are going to hear that piano elsewhere in the house!
As to the player feeling uncomfortable with the door open or closed or what have you, that depends on how the others in the house treat the player, not on the presence or absence of a door.