What is the pressure belonging to in Boyle's Law?
Your derivation of Dalton's Law of partial pressure is fine. 

I think you are confused about how pressure is "applied" to a gas, and what that does to the internal pressure of a gas. 

Imagine that you have a "sphere" of gas surrounded by a "force field" (like in Star Trek) in empty space, then the force field containing the gas is dropped. Since there is nothing to confine the gas it just expands into infinite space. 

A lot of ideal gas models, like the Wikipedia article on Boyle's Law, show the gas as being contained in a cylinder with a piston pushing on the gas to create the volume and pressure of the system. The external force on the piston is counterbalanced by the pressure of the gas times the surface area of the piston. The pressure of the gas is the same on all surfaces of the container. The gas doesn't just "press back" on the piston. 

Pressure and volume are inversely related for the same system. So at the same temperature, 

$PV = k$

Thus if the pressure increases, then for the same amount of gas the volume must decrease to keep $k$ a constant.