If all of the water in the Caspian Sea were poured over the summit of Mount Everest, would it freeze, and how much higher would it make the mountain?

That amount of water would probably heat up the mountain and wash away a fair portion of it.

If the water was slowly put on the mountain cold enough to be snow or ice it would continue to roll/slide down the mountain to melt at lower elevations. The height would not increase by much. Maybe 50 to a couple hundred feet.

If the water was somehow retained and frozen in place… The Capsian Sea is much larger in volume than Mount Everest, so let’s imagine three scenarios.

1. Very wide block of ice. The Caspian Sea is considered to contain 19,000 cubic miles of water, Everest is something like 5.5 miles high. As such a block of ice 5.5 miles high with the volume of the Caspian Sea would cover about 3,500 square miles.
2. Roughly cubic block of ice. Cube root of 19,000 puts us at about 26.7 miles on each side. That about 140,000 feet high. Commercial jets fly at about 30,000 feet. Most ice couldn’t take this configuration and would crumble.
3. Very tall block of ice. With this scenario, we could easily get to space which roughly starts at 82 miles up. This would still be 230 square miles across. Again this would likely quickly break up due to the stress imposed on the ice.