If I drop a leaf twice from the height of a tree in a completely controlled environment, will the trajectory in each case be the same? Putting my question in other words, can earth form again if a similar initial universe condition is given? The uncertainty principle says that we cannot tell with certainty the position of a particle if we know its velocity with greater surety, and vice versa. But I have always felt that this restriction will vanish for a 'god' who has the advantage of knowing how the particles initially were in the beginning, and how they would interact and how the story would go on.... Therefore, can complete knowledge of the initial conditions of the world completely remove the uncertainty principle?
 A: No. The Uncertainity Principle states the following: 
The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. There is a minimum for the product of the uncertainties of these two measurements. There is likewise a minimum for the product of the uncertainties of the energy and time. $$\Delta x \Delta p \geq \frac{h}{4\pi}$$ 
This does not mean that these parameters are impossible to measure because our instruments are not accurate enough yet or maybe some error creeps in. Doesn't matter what the initial conditions are, this uncertainity lies in the very nature of matter.
The main question that you are asking is not really affected by the Uncertainity Principle. The Universe is not deterministic. Hence, the formation of our planet having that tree and that tree having that leaf are just what you may call accidents. Our version of the Universe is just one out of an infinite others where there is no Earth or us.
A: The problem with this question is that, even if you perfectly controlled the conditions at the macroscopic level -- including somehow releasing the leaf in exactly the same way every time (nearly impossible), and using leaves in identical starting state (the same leaf may have lost some water by the time you repeat the experiment, and suppressing all pressure waves in the enclosure (absolute silence)... there are going to be feedback effects at the microscopic level which the experiment amplifies and makes visible. Brownian motion could be enough to affect air density -- or leaf flexibility -- enough to start a divergence, and that divergence will affect the leaf's path, which will affect the pressures on it, which will affect how it curls, which will affect its path, which...
This is a highly unstable system, and in practical terms it's probably impossible to control it well enough to exactly reproduce the experiment.
None of which has anything to do with the uncertainty principle.
A: In the case of a leaf's trajectory, yes, it'd be same in both cases if the environment is exactly the same. And you can generalize this to any macroscopic event.
As for the formation of Earth simulating the universe since its beginning, it's complex because it's not a macroscopic event. In the quantum world, everything is probability driven. In a heap of radioactive atoms, all atoms are equally similar and equally unstable. Yet some atoms bleed out their instability at one point, and some after billions of years. See, I haven't even used the uncertainty principle here.
So, given the initial conditions of the universe, the formation of Earth is one of countless possibilities.
A: You are misunderstanding the Uncertainty Principle. The Uncertainty Principle says that a particle cannot simultaneously have a definite momentum and a definite position. This is not due to our incomplete knowledge of parameters. This is a fundamental law of the universe and arises from the fact that the momentum and position operators do not commute in Quantum Mechanics. In your question, the leaf outcome will be the same in both experiments because when you combine a lot of particle wavefunctions together the object does not behave probabilistically anymore. This phenomenon is called Quantum Decoherence. In the earth formation experiment, we simply cannot know. You need the right conditions for the earth to form, and because the Uncertainty Principle and Quantum fluctuations did play an important role in the early universe conditions we can't know for sure if its going to happen again.
A: Different trajectory each time due to probabilistic nature inherent in quantum mechanics and the uncertainty principle. The uncertainty principle is not "removable", it is not a constraint based on practical limitations in an experiment, but inherent to quantum mechanics calculations.
There was an interesting thought experiment, given absolute control over a much simpler system (multiple collisions between 12 billiard balls), it is not theoretically possible to determine the trajectory of the final object: D J Raymond - How Determinate is the "Billiard Ball Universe"?
How different the trajectory ends up will depend on how important this microscopic randomness becomes. I would expect the equations of motion for a floating leaf to have some non-linear elements, such that tiny amounts of difference in speed or position will lead to essentially chaotic motion. 
However, that may be constrained within a narrow bound. I cannot really guess whether the chaotic parts of the motion would be one part in a million or completely overshadow the regular predictable motion of the leaf (that you might get from a classical fluid dynamics solution to the equations of motion and fluid flow around the falling object). 
Some leaf shapes and airflow may in fact be relatively stable and not show much difference (trivially, with very low air density, the trajectory will be a straight drop). But I suspect that the iconic image of a leaf spinning and rocking as it slowly falls is a highly chaotic system, and it will then be vulnerable to effects of the uncertainty principle, assuming your imagined "perfect" experiment set up is done within the bounds of known physics.
A: From my understanding of your questions, you are confusing the "scientific method" and the "uncertainty principle.  The scientific method says that "given the same starting conditions, within a controlled environment, etc., the "results" should be the same (ei. repeatable within some degree of accuracy).  The uncertainty principle "deals" with an entirely different thing.  It tells us that either one of the measurements of a particle's position and momentum can be accurately determined, but not both at the same time. 
