Let me begin by saying how much I have enjoyed your "Song of Fire and Ice" series, and how much I am looking forward to the succeeding volumes.
Since I have two degrees in Geography and I work with maps, whenever I read a book, I am obsessed with the spatial relationships between cities/locations/continents/etc. I have a few geographic questions for you, if you would be so kind as to answer (although I understand you not answering if an answer would give away a plot detail).
1) Are we ever going to see a full map of your world, showing Westeros, the Summer Sea, the Jade Sea, etc?
There will be new maps in each book, so the pieces will gradually come together... but I am not sure that I will ever do a map of the entire world.
I like the idea of far off realms being fabulous places, and the maps getting vaguer out toward the edges... as they did in real life. "Here There Be Dragons," and all that.
2) What is the approximate distance between Westeros and Valyria (shown in the map labeled The Lands of the Summer Sea). Also, what general direction would you head if you travelled from Westeros to Valyria?
To get to Valyria from Westeros you would likely have to go south a bit, then hang a left and head east.
3) Along those same lines, how far is it from Valyria to the 9 Free cities? Are the 9 Free Cities located in the general area of Qarth on the map "The Lands of the Summer Sea"?
No, the Free Cities are much closer to Westeros than to Qarth. Several of them are just across the narrow sea -- Braavos, Pentos, Myr, Tyrosh, Lys.
The others are a bit further east.
4) What did the Broken Arm of Dorn connect to? Was it the eastern continent? Did it connect somewhere around Sothoryos?
It was a land bridge that connected Westeros to the eastern continent.
Sothoryos is the southern continent, roughly equivilent to Africa, jungly, plague-ridden, and largely unexplored.
5) Does Westeros connect to the eastern continent through the north?
I hope that helps. Keep reading.