Collision that governs kinetic model of gas From Atkins' Physical Chemistry 10th, it states that

The kinetic model is based on three assumptions:

The gas consists of molecules of mass $m$ in ceaseless random motion obeying the laws of classical mechanics.
The size of the molecules is negligible, in the sense that their diameters are much smaller than the average distance travelled between collisions.
The molecules interact only through brief elastic collisions.


I quite can't get the second assumption. The intermolecular distance between the molecules are greater than the diameter of the molecules, so we can neglect the volume. Given Boyle's law apply (as omitting size and volume of molecules is a property that ideal gas has), there will be many molecules collide with the walls and the molecules will hit another molecules after its collision with wall (right?), thus the pressure of the system should be low. How often the collision of what kind happens under what condition?
If the separation doesn't result any influence on each molecule (because third assumption said elastic collision, so any interaction (hydrogen bonding, van der Waals, so on) higher than the collision energy should be omitted), will the collision come purely from their collision with walls, not collision between each molecule?
Wikipedia states also the assumption that

The number of molecules is so large that statistical treatment can be applied.
There will be many molecules collide with the walls and the molecules will hit another molecules after its collision with wall (right?), thus the pressure of the system should be low. How often the collision of what kind happens under what condition?


The collision condition doesn't need to be strictly in Boyle's law condition, that the elastic collision is happened between anything and under any pressure condition. The requirement of large number of molecules but only collision interactions are applied implies that it is just not applicable to apply Boyle's law in kinetic model of gas.