Why the volume of a cell in phase space should be equal to $(2pi hbar)^s$?

We want to properly define the concept of entropy using the Boltzmann’s Definition of it. But there is a big problem: the coarse graining problem (Id est: How do we count the number of microstates in the allowed region of phase space? By dividing the phase space in cells? Good idea! But how big should the cells be? )
Fortunately Quantum Mechanics comes in our help: if we consider, for example, the case of our system being an ideal gas, then we can show, apparently, that the volume of a cell in an $s$-dimentional phase space should be $(2pi hbar)^s$. (Reference: Landau, Theorical Physics Volume 5, page 40)
This fact should arise, I think, from considering the motion of a free particle confined in an $n$-dimentional box; the energy states of the particle, as well as its momentum, then should be quantized, we can then use this fact to state that the volume of a cell in phase space should be big enough to include into itself only one of the possibile quantized states. In this way the choice of the volume of the cell is not arbitrary, and we can derive it exactly.

However I have three big problems regarding this approach:

1. For a particle in a box energy and momentum are indeed quantized, but the position is not; phase space should be composed of position and momentum, but only one of this two is quantized, so seems impossibile to me to use quantization to determine what volume the cell should have.
2. If somehow we can solve my first problem, then what mathematical derivation tells us that the volume necessary to have only one quantum state in a cell is $(2pi hbar)^s$?
3. All this reasoning is based upon the hypotesis of dealing with an ideal gas, so no interaction and we can consider, for the derivation of the volume of the cell, a free particle in a box with its simple quantization of momentum; but what if we are dealing with something else? Should we change our value $(2pihbar)^s$ for the volume or there is some trick that allows us to keep using it even when not dealing with an ideal gas?