Well, in most cases it is what happens, the checkmated King is really taken otherwise the attacking player is letting a new Prince that could become a King again, which is not suitable.
Not really. The new Prince is not any different from a Prince that has always been a Prince, and whether you want to capture a Prince (or in fact any piece) depends on whether it is protected, and how much you would lose on the recapture. This is quite similar to extinction royalty (except that the checkmated side in Tamerlane II has to spend a move on the succession, while with extinction royalty the remaining King automatically will become a King that has to take threats seriously).
For instance, in Spartan Chess black starts with two Kings. Now the tactical value of a King is not very large; it is close to a Knight. The fact that you have a spare royal is also worth some, though. In playtesting it turned out that when replacing a Rook by an extra King for one player, that player will lose more often than not, about half as bad as when he is a Pawn behind. So a Rook is about half a Pawn better than a spare King, and you would not want to trade a Rook for that King. Let alone a Queen.
Not really. The new Prince is not any different from a Prince that has always been a Prince, and whether you want to capture a Prince (or in fact any piece) depends on whether it is protected, and how much you would lose on the recapture. This is quite similar to extinction royalty (except that the checkmated side in Tamerlane II has to spend a move on the succession, while with extinction royalty the remaining King automatically will become a King that has to take threats seriously).
For instance, in Spartan Chess black starts with two Kings. Now the tactical value of a King is not very large; it is close to a Knight. The fact that you have a spare royal is also worth some, though. In playtesting it turned out that when replacing a Rook by an extra King for one player, that player will lose more often than not, about half as bad as when he is a Pawn behind. So a Rook is about half a Pawn better than a spare King, and you would not want to trade a Rook for that King. Let alone a Queen.