Is it permissible for me to sit with my fiancé in a room with an open door in the presence of my family, and is it permissible to communicate with him by phone to ask about studies or work, or to congratulate him on Eid, or for matters of marriage?

The fiancée is a stranger to her suitor, and it is not permissible to speak with her except out of necessity. Necessity is defined as that which, if left undone, would cause a person hardship and difficulty. Contacting her to congratulate her or to ask about her study results is not considered a necessity, and therefore, it is not permissible.