Is it permissible for the fiancé to be alone with his fiancée, provided that the room door remains open and with the knowledge and consent of both parents?

It is forbidden to be alone with one's fiancée if the marriage contract has not been concluded, because she is still a non-mahram (definition). It is not permissible to see her except out of necessity in the initial stages, without seclusion or repeated meetings. The guardian of the girl must prevent the suitor from entering upon her and sitting with her alone.