When you haul something it is usually heavy, basically it's something you need to use force with. For example
I had to haul my car into the garage (obviously we don't say carry because a car is too heavy to pick up)
Carry is something you actually pick up, for example 'I carry a cat' because it's something you can easily hold in your arms.
Removal is the act of removing something, like 'I removed the engine of my car and now it won't start' (due to the fact you pulled out the engine)
Flitting is the act of moving fast and lightly~ 'The monkey was flitting from tree to tree'
Summary:
Haul = pull with force (usually said when you have to drag something heavy)
Carry = something you can actually hold when you move it from place to place
Removal = The act of removing something
Flitting = Moving fast and lightly from one place to another
Hope that helped~