You choose the direct ip connection then copy and paste "caledorn.no-ip.org:2060" and put that to the network address and connect. That worked for me.
Yup, you got it.
The turn file is never sent around. Game is hosted on the server at all times, and we are all playing roughly simultaneously.
The civstats site is just to track who has played the current turn, who hasn't, time remaining, etc. And it emails you when a new turn is up. Since it lets you see who has played and who hasn't, that is how you keep track of turn split once you get in a war with someone.
So to re-iterate.
Launch Civ4BTS
Pick Multiplayer
Pick Direct IP
put your forum name in the "name" box
Ignore the whole "host game" side of the box.
On the Join Game side
Put "caledorn.no-ip.org:2060" in the network address box
leave the password blank
Hit connect.
List of civs will come up
Pick yours by nation (verify the correct leader pops up)
Hit OK
SET A PASSWORD (later you'll just enter this same password)
Play your turn
Hit end turn (when you do, this information gets handed off to the Civstats website, and puts a star by your name until the turn flips.)
Exit to main menu (please don't pick "exit to desktop", that sometimes causes the server to lock up...)
If you are the last player to play the turn, when you hit "end turn" the server will advance the turn while you are logged in. You can play your next turn immediately (don't do this during a war..). Hit end turn again..
If the time runs out while you are in-game, the server will flip the turn.
If the time runs out, even if you haven't logged in, the server will flip the turn. In that case it will take any actions you had queued up for you.