Here’s my question. My Orthodox friend is emotionally involved with a man who is seeking to convert. Is it halachically permissible for her to continue communicating with him during his process? I ask because my understanding was that they have to stop communication, to insure the conversion is Halachically alright.

It is true that a convert has to have pure motivation to convert and a Jewish girlfriend is not that pure. However if it is clear that he will convert regardless to the relationship he holds with her, it might not be essential for her to leave him. This can only be discussed with the Beth Din that is in charge of the conversion and is familiar with the details of this case first hand.