Is there a way to derive the formulas for point addition and multiplication on elliptic curves in Jacobian format by yourself? How could I have derived these formulas by myself?
I haven't worked through all the (boring) details but it should be fairly easy. Just start with the normal affine addition and doubling formulae but when it comes to the stage where you do any division, don't bother and just keep track of the numerator/denominator separately. The whole point of Jacobian and other projective representations is that you don't do any division until you HAVE to. The disadvantage is that one point has many different projective representations.
The $Z^2$ or $Z^3$ aspect is just shuffling multiplications around to make doubling slightly faster versus addition or vice-versa or to do with being able to add an affine point to a projective point more quickly. It tends not to matter too much. If you want faster calculations you should be choosing perhaps an Edwards curve representation and/or maths over an appropriately fast extension field.