I disagree with the multiple personality theory.
Multiple Personality Disorder (aka Dissociative Identity Disorder) tends to manifest itself in slightly different ways than what was happening with Luna. The actual disorder is often associated with other symptoms such as depression, memory loss, and panic attacks, none of which I’ve seen with Luna.
What seems to be happening with her is a more fundamental change, probably due to (I hate to say it) magic. Explaining the difference between Nightmare Moon and her “issues” and “royal” forms in that way is easy enough. But, explaining the difference between the “issues” and “royal” forms is more problematic. I tend to think that, instead of a radical personality shift, it was a logical progression, a normal personality change. Her development into Nightmare Moon took a long time, and she was reverted to “issues” Luna by the Elements. Then, her transition from “issues” Luna back to “Royal” Luna took time as well.
Her Nightmare Moon form was clearly an expression of her resentment for Celestia. Her “issues” form represented the internal conflict between the darkness that she became on the Moon and her origins as Celestia’s little sister. This is a more child-like form (as evidenced by the non-flowing mane, etc) than her previous, “royal” form. Over time that was resolved, and she reverted to her “royal” form, which I tend to think she was before being banished.
Now as for explaining the physical changes, I think that they are psychosomatic, and since magic is biologically part of alicorns, it would be relatively easy for her appearance to change.
In summary, I don’t think Luna has multiple personalities, I think we’ve witnessed her personality logically changing from Nightmare Moon to “issues” to “royal.”
About the “falling in love with question,” that’s not something I think about, but I’d say Futashy.
In actuality, I think Twilight, mostly because she is my favourite – nice, smart, etc, but I don’t really want to think about getting romantically involved with ponies.