I could say i'm in stable relationship with my girl, despite the fact that we both currently live like on the other sides of the world. She's okay with it, since i promised i marry her and let her move to live with me here where i am next year. As for keeping contact; i try to call her everyday, but given the current situation of life i'm in, i'm not able to do that as often i would like to.
I know for most people i might sound crazy for having gf so far away i can't see her daily, or even weekly ;)

The point behind me telling all this is that if you two come along well, i see no problem with having only that sort of distance between you. It might be difficult to convince her that it's not a problem for you, but you could try to call her regularly and see if that would make her believe that you're serious.

And what to do on date? When we got together it was really easy for us to do something, as i was just a tourist on a whole new continent for me. Just walking around and seeing many new things with her was awesome. But as you seem to live in same city, i think this is kinda out of option.
But i'd say that just walking around will eventually get you somewhere. I mean just think about place to start on your date, like marketplace or something "unusual" and i'm sure that from there it's easy to go from place to place.

Just my two cents, and some of my own experience here. But i really think that when you really love someone, and when someone really loves you, you need nothing to to besides be together...