For an object falling in a fluid, pressure on that object $P=\rho\cdot g\cdot h$ Is $h$ a function of time or depth? i.e- does pressure change to time or depth?
$P=\rho g h$ is not a function of time or depth as much as it is a relationship that holds in all circumstances (or at least all circumstances within the domain of the model). If you are in any situation with fluid pressure, it is expected that the static pressure at a point will be exactly equal to the density of the liquid times the acceleration of gravity times the height of the column of fluid above the point.
If you have a situation where something is changing, then you can use that relationship to create a functional relationship. That could be $P(t)=\rho g h(t)$, in which case pressure and height are functions of t. It could also be $P(t)=\rho g(t) h$, which seems a bit absurd, but if you looked at the pressure at the bottom of a container on a rocket ship and that rocket ship flew far away from earth, the varying components may be the acceleration of gravity (which decreases) and pressure (which also decreases), while the height of the column is a constant.
You could also come up with some interesting cases in brackish estuaries where salty water meets fresh water, where the most meaningful functions could be functions of distance from the mouth of the river, rather than time. $P(x)=\rho(x)gh$ might be very meaningful if the density of the water is changing as a function of distance.
The formula you mention is the Stevino's law, which tells you the pressure of a column of fluid on an object on the bottom of the column itself. If the object is in the middle of the fluid, then you should consider also the pressure from all the other sides, i.e. Archimedes' principle: the force acting on the object depends essentially on the comparison between the object's density and the fluid's one. So, if an object is falling (and if this is not happening because of the initial inertia of the body) then it will fall until reaching the bottom. I hope this can help you.