I was wondering if the viscosity force depends on the depth of water. I mean it is much more harder to swim in the ocean at a depth of 1km than swimming at the surface due to the viscosity forces ?
The viscous force on an object due to a fluid is directly proportional to the velocity gradient of the fluid. So at great depth, the velocity of the fluid is not the same as that at the free surface of the fluid. Hence the viscous forces are also different at different heights. Use your mathematical intuition following these ideas to come up with an expression for the relation between viscous force and height of water.
Probably not, assuming you'd survive the pressure. This paper shows that the viscosity decreases by a small amount with pressure, at most about 10% for a pressure at the bottom of the ocean, which is small compared to changes in viscosity with temperature. Also, the density of water is only a few percent higher at the bottom of the ocean. So the conditions that contribute to the forces you'd feel swimming near the surface are only slightly different, drag-wise, than those in deep water.