1. For the pendulum system, the equation of motion is

math_failure (math_unknown_error): \frac{d^2 \theta}{dt^2}= -\frac{\ell}{g} \sin\theta  
The motion of the pendumum will be the same if the water always leak out in the radial direction
(Did not change momentum in tangential direction).
However, the pendumum motion will be changed if the water leak out has momentum in the tangential direction of the pendulum motion.
2. Does the mass always doubled every time it reach maximum length or just doubled once?

You can guess what will happen with the following analysis:
The potential energy for a spring is U(x)=\frac{1}{2}k x^2, where x is the displacement.
All the  potential energy will convert to kinetic energy when it reach the equilibrium position.
i.e. \frac{1}{2}k x^2 =\frac{1}{2} m v^2.
If the mass is doubled, then the velocity will become smaller v'=v/\sqrt{2},
and the oscillation frequency will be smaller too. \omega= \sqrt{k/m}