Why are the south, the west and the north of the Caspian sea lush but its east completely dry?
The mountains.

The south has a steep relief, as well as the west. That means that there is much precipitation. The clouds throw every drop on these mountains, so almost nothing is left. The east is pretty low, so the remaining clouds (not much left) just fly over. In fact, deserts are generally not steep and they are usually behind the mountains. That is right what happens here. On the north, the effects of the west are still felt, so it isn't so dry, but it is pretty low, too. Also, I wouldn't say that it's lush : )