1. Yes, ask for the spin in the third trigger pull. If you don't have me spin again before the third pull of the trigger, you have a 1 in 3 chance of losing the vase on that shot--which is the same as if you do ask me to spin. But the advantage of asking for the spin anyway is that if I spin first and the vase survives the third trigger pull, then you will return to the 1-in-4 chance of losing the vase on the fourth trigger pull.
If you use that strategy (don't spin before the second trigger pull, spin before the third, and don't spin before the fourth) following the first click, your vase's chances of survival for the last three clicks are: 3/4 * 2/3 * 3/4 = 3/8. If you had asked me to spin every time after the first click, the probability of the vases survival would have been only 8/27, a little less than 1/3.
2. If there is only one bullet in the chambers, then you would want me to spin every time, so every click leaves that vase a 5/6 chance of survival. Failing to spin will give a lesser chance of survival.