Can a submerged object in a free falling fluid sink faster than falling?

I want to supplement Emilio's intuitive answer discussing what would happen with some thoughts as to why what you propose in your second part cannot happen.

What kind of object could fall out of the water and reach the ground before the water does?

Let's assume the water is a single entity. In order for the object to accelerate faster than the water, the object needs to have a larger downward acceleration than the water does. This would need to come from a downward net force that is larger than the weight of the object, as in free fall both objects will have the same downward acceleration. Where would this force come from? There is no buoyant force in a free-falling fluid, but even if there were the buoyant force would act upwards on the object, not downwards. Therefore, the best your could even hope for is that the object and the water move together, and this is indeed what happens.

Perhaps the confusion comes from why certain objects normally sink. They aren't pulled down by the water, they are pulled down by gravity. In your everyday scenarios it is just that the fluid impedes this "falling". You could even think of us as all sinking in the Earth's atmosphere. Therefore, in your scenario, it is not the case that because the object is in water suddenly means it wants to move down through the water. The water itself is not the mechanism for why objects sink.

The Equivalence Principle tells you that physics in free-fall is identical to physics in an inertial frame in the absence of gravity, in which case there is no buoyancy and any changes to the boundary of the liquid come from surface tension and related effects. The same is true for your falling mass of water.

under what conditions could the object fall first?

Air resistance.

If there is enough air resistance to slow down the water, then the denser than water object will fall faster than the water.

Except OP said the liquid was free falling, so air resistance doesn't apply here

flip the ocean, how does the pressure affect things

This is quite different.

For one thing, it's not free-falling.

The high pressure water that is now at the top will expand - some of the water at the top will initialy be pushed up, possibly with enough force to lift the heavier-than-water object. It will then fall. Water not at the top will fall from the instant the sea is 'flipped', though perhaps sink is a better word because this isn't free-fall, it's compression (at the bottom) and it's decompression (at the top).

There would be more than a bit of bouncing and turbulence, but before too long, the original pressure distribution will be restored.

Meanwhile, possibly after having initially been thrown upwards, the heavier than water object will again sink to the bottom. Depending on friction, the object might drag down some water with it, but most of the water at the top will stay more or less the top, and most of the water at the bottom will stay somewhere near the bottom - until temperature convection and other causes of turbulence come into play. But I think that's going beyond the scope of the original question.