Pressure in PSI is not the only factor, when it comes to temperature based effects. The material(s) the football is constructed of change properties considerably with temperature. Hardness, stiffness, and likely internal damping of the materials all increase with decreasing temperature, which would tend to (mostly) counteract the pressure drop. (The damping change might add to the pressure drop in creating a less “bouncy” ball.) I would not be surprised if “operationally” those “materials” factors are more significant than the effect of the pressure drop.
Yup.
As you say, the leather surface of the football becomes harder (not softer, despite the pressure change). In this state, the football is often described as a rock, as it becomes relatively non-bouncy.
Water has multiple effects, including affecting weight and balance.