Alone in the room together?

In that situation it would be allowed, for a few reasons. He would be considered too old, and probably married, so it would be like being alone with her father or uncle (which was allowed). As for the nightgown, pay attention to what she's wearing when she's in the bed versus when she's actually talking to him. She throws on a quick dressing gown which takes care of propriety for her being in her underthings.

Another point about that scene is that his message is supposed to be a secret one, so any people who might have been in the room (ladies in waiting) would have been sent out anyways. Any risks with her reputation would have been quickly solved since she wasn't supposed to be in the kingdom after the meeting.