You've got some stuff mixed up. You can't just "run the map", as you put it, through Hammer until it's already compiled. You're getting confused because of the Run Map dialogue box, which compiles the map and launches it in game if you wish. They aren't two separate processes. Expert mode is accessed through this compile/run map dialogue box, there's a little button on there that says "Expert" and switches it expert mode, or "Normal" to switch back. You might already be in expert mode.
BSPs are created when you compile. The BSP is the product of the compile process, it is the map itself. If you've already played any of your maps, then you have already compiled them.
I'm thinking you should read all the beginner tutorials to understand how this process works, and clear up the confusion.