He is in the confession dial, which contains the castle. "Bigger on the inside", just like any other Timelord tech. The confession dial probably was in Gallifrey… we have to wait for next week's ep for how it happened.

He IS in the lake. If you are talking about the scene where he was in the TARDIS, that was in his mind.

It's explained in the episode. The concept of the teleporter, as have been discussed in many other sci-fi shows like Star Trek, is that it isn't really a teleporter ("Teleporter - fancy word, just like 3D printers, really.") that transmits a person from one location to another. Instead, it atomizes the original person…