Yes the Koppen classification mainly use these two to classify the climates. But others might use vegetation or something else to classify.
But rainfall depend on wind direction. this mean that some climates always appear at specific places. For example: Mediterranean climates are always on the west coast near a body of water.

Alpine tundra is still a tundra: The temperature drop by about 5,5°C for each 1000m. So if you consider that the average yearly temperature at the equator is around 25 to 35 °C then altitude would be around 4500m and 6400m or maybe more than that depending on the surroundings. But if it's much higher than that the climate become too cold to be considered a tundra and become land of eternal ice.