Why is the sum of two inexact differentials exact?
There's not much to do, as you nearly spotted the difference by yourself: $U$ is a state function, thus is path-independent. In other terms, one can estimate it at any moment only with an initial and a final value (and express it in terms of its variables only). But with $Q$ and $W$, it's different: these are path functions (or process functions) consequently depending on the path. Hence, their differential is not something one might express depending only of its variables. (these are not &quot;real&quot; functions for mathematicians...)

Is the sum of two path-dependent functions a path-independent function?

The sum of inexact differentials is an exact differential in that particular case. In other cases, it depends on the units involved: for instance, with entropy, we have $\mathrm{d}S = \delta Q_\mathrm{rev} / T$. So, no typical case here!

what is the difference between a partial derivative and an inexact derivative?

These are absolutely not the same! A partial derivative is a differentiation of a multi-variable function regarding one of its variables only, while what we have here is an inexact differential, thus not a derivative as 1) no variable is specified here, and 2) an inexact differential can be of a function depending on only 1 variable.