Do depth or amount of water affect the lifting force? Lifting force depends on the volume in water and the density of water. 
Is there any possibility that depth or amount of water can affect the force, in real life conditions?
 A: 
Is there any possibility that depth or amount of water can affect the
  force, in real life conditions?

As you said, the lifting force depends on the volume of the displaced water and the density of the water. So, if the volume of a submerged body does not change, but the density of the water does, the lifting force should change as well.
Can density change under real life conditions?
Although water does not compress much, only about $46ppm$ per $1 atm$ of pressure, it adds up. Assuming sea water density of $1025kg/m^3$, the pressure increase at $4000m$ depth will be $\rho gh=1025*9.8*4000=40180000 Pa$ or about $396 atm$, which translates to $46\times10^{-6}\times396=0.018$ or $1.8\%$ shrinkage of volume or $1.8\%$ increase in density (thanks to Rafik Fayzulin for pointing the significance of that factor). 
In addition, water density is affected by temperature and salinity: it decreases with temperature and increases with salinity.
The picture below (copied form this article) shows a typical curve of density vs depth in an ocean:

The article explains how temperature and salinity affect density at different depth zones.
So, the density of the water does change with depth and therefore the lifting force will change, too.
