On the romantic issue : captains with cabin crew, medical doctors with nurses,....so why not instructors with students. We all are only human so if a spark ignites.... But if this would happen, I would immediately no longer be instructing this individual.
On the friendship/social issue : in my company we are all for a long time far away from friends, family and comforts of home. So yes, we do meet, students and IP's in the few restaurants or clubs. And these encounters are also a channel to get feedback from them about their moods, feelings, etc... that you would not observe on the training base.