I have read that chariots were first used in mesopotamia around 3000 BC. Did all war chariots used later derive from that first use, or were they invented independently in different places? Did all war chariots use horses for the motive power, or did someone try other animals, such as dogs, reindeer, or elephants?
The answer to this question really depends on how one defines chariot.
If we consider a chariot to be any type of wheeled platform pulled by animals, and is intended to be used in battle, then our earliest evidence comes from around 2500 BC. It is called the Standard of Ur, and shows a Sumerian army:
https://www.britishmuseum.org/collection/object/W_1928-1010-3
The Sumerians themselves were a civilization originating from what is now Southern Iraq and Kuwait. The carts/wagons themselves appear to contain a quiver storing javelins, and if this is the case one could argue that such carts/wagons were not intended to charge infantry directly, but rather function more in the capacity of skirmishing and harassment.
Now, if one thinks that a chariot can only be called as such if they have two wheels, then our earliest evidence is from what is called the Sintashta Culture in Central Asia. The Sintashta Culture is believed to be Indo-Aryan in origin, and remains of chariots have been discovered dating back to 2000 BC and earlier. It only after that point that what we could consider the 'classic' chariot starts to appear in the Near East, India, and China.
Sources
Shadows in the Desert: Ancient Persia at War, by Kaveh Farrokh
Warfare in the Ancient Near East to 1600 BC, by William J Hamblin