1&2: physics.
caustics are shifted because the light from sun is refracted. This means it changes direction and hits the surface in different place that it would on its own, forming the caustics and shadows, such as here:
The reason your results look unnatural is because your setup is unnatural - your media are not correct, some of the rays hit the floor while in air and other hit the same place while in water. To correct this you would either have to model the water twosided, or make the pool water enclosed, like in reality. Then you would no longer get such effect.
The caustics visible through water are not missing, it just has MUCH higher variance, because it represents an SDS (specular-diffuse-specular) light path, which is extremely difficult for BDPT to render. You have few fireflies there, these fireflies are so bright, that they "contain the entire caustics intensity from surrounding areas" (simplified), so it is unbiased in the end (no energy is missing). To render it efficiently you would need different light connection strategy, like progressive photon mapping, or VCM (which is BDPT + progressive photon mapping combined). Try it and you will se the missing caustics.
