They do basically mean the same thing, as Randy and Jim have said, but I always find it more common that dating someone means they are in a relationship, as in boyfriend/girlfriend, and seeing someone doesn't mean the people are actually dating. But because they mean the same thing, it's not uncommon to hear people use them either way. Usually they will say more about what type of relationship it is anyway. I hope that helps :D

They are the same. Note that the multiple meanings of "seeing" can cause ambiguity: "She is seeing someone who is very handsome." This probably means that she is dating someone handsome, but it could mean literally at that moment that she is looking at someone who happens to be handsome.

The terms mean the same thing. If we say two people are dating, that suggests a habitual, ongoing process, in which case I would say they have a relationship. The word "relationship" is a neutral word that suggests nothing else necessarily.