How is it possible, and what does it imply, when we say that the sum of two inexact differentials is an exact differential?
In the example at hand it means that we divide the ways we can change the energy of the system into exactly two bins. The first we call heat, and all the other ones we call work.
Heat is an energy transfer that occurs when two entities of different temperature are in contact. The difference in average kinetic energy of the particles
causes an energy transfer during collisions which overall increases the temperature of the cooler one and decreases the temperature of the hotter one.
Work is everything else, from compression of a gas to stirring to powering a heating coil inside the system with an electrical power source outside of the system.
We can have different amount of heat and work depending how we run a process. If the starting and final state match for two processes, the change of the system's energy will be the same, no matter what path we choose (i.e. what the individual contributions are).
Is the sum of two path-dependent functions always a path-independent function?
We could have divided up the work into pV-work and non-pV-work. Both would be inexact differentials. If we add them up, we would still have an inexact differential.
So the answer to your question is no.
Also, what is the difference between a partial derivative and an inexact derivative / differential, apart from mathematical implications?
A partial derivative is when you form the derivative of a multi-variable function with respect to a single variable. So if my function T(r) describes the temperature distribution in 3D space, and I form the derivative with respect to the height $z$, that is a partial derivative. This is different from the total derivative, which would tell me the gradient of the temperature at a given point (i.e. in which direction it changes most). So a partial derivative and an inexact differential are quite distinct.