Why does pressure increase with water depth?

The problem is that you're modeling the liquid like an ideal gas, whose molecules independently bounce off the ball, but liquids are characterized by strong interactions at short distances.

A better (but still inaccurate) model would be to treat the liquid like a solid locally, i.e. imagine each of the liquid molecules connected in a chain by springs. An increase in pressure means that the springs are compressed more and more, so they push outward onto your object more and more.

In terms of your variables, we should have $F \sim k \Delta x$, not $F \sim 2mv/\Delta t$. In this model, pressure can be transmitted from molecules far away, just like tension is transmitted through a rope.

Pressure is higher because at deep water you have so much more water on top pushing down on you because of gravity.

Think of water molecules at a certain horizontal level.

They have a weight and forces acting on them from the water molecules above both of which must be balanced by an equal upward force from the water molecules below.

Those water molecules below exert a larger force on the horizontal layer of water molecules because their average separation is slightly less that the average separation between the molecules above the horizontal layer of water molecules.

So think about the interactive forces between the water molecules rather than worry about the average separation between the molecules.
The temperature being lower would mean that the average separation of the water molecules will be less but the idea of balancing forces still stands.