AFAIK, pieces can move through the Throne to the opposing side as long as it's uncaptured. After capturing it, can the opposing player still move to the other side? What about the controlling player?

Thanks! :)

after it's captured, nobody can move through the Throne.

Another question; regardless of which player is unable to move, when either player is out of legal moves the game ends and the winner is the current controller of the Throne, right?

Yes that is right and if nobody has the Throne, then it is a draw.

A small but crucial clarification.  When either player - on his turn - is out of legal moves.  In other words, after I've just made my last possible move (and am therefore out of them), you have one more turn at your disposal - and then the game ends.