Usually, if you increase the K-point mesh the calculation becomes more accurate but becomes more and more expensive. The same is true for cut-off energy. One should test and converge the K-point and cut-off energy with respect to the total energy of the system.
The size of the supercell depends on the user's choice of what kind of cell dimension with which the user wants to start the calculation. In the periodic boundary condition anyway, the periodicity will be maintained according to Bloch's formula. A large supercell with a smaller K-point calculation is equivalent to a small supercell with a large K-point calculation.
If you increase the size of the supercell while keeping the K points fixed, of course, the absolute total energy will decrease as you are now increasing the number of particles. So more negative energies are expected.