Dubai Mall, Burj Khalifa and the Aquarium are all one place. The Aquarium is inside the Mall which is adjacent to Khalifa. The cost by taxi from the airport would be around 20aed + 20aed fee which is applied to all taxis leaving from the airport. Alternatively you could ride the metro from T3 to Dubai Mall which is only a few dhs (sorry i'm not familiar with their pricing). The travelling time by taxi should be no more than 20mins.

I dont really see much purpose in going to the Palm. Its impressive when you see it from the air but when you're on it, you might aswell be anywhere else because its a flat surface with mostly residential blocks. At the very top end of the palm is Atlantis Hotel which has some restaurants and also a small walk thru aquarium but in my opinion, i would waste my time (and money) going all that way.