Is it an open problem if generalized resolution is always possible?
Resolution is refutation-complete. This means that if $S$ is a set of unsatisfiable clauses, then you can use resolution to derive the empty clause. Moreover, Resolution is sound, so you can only derive the empty clause if $S$ is unsatisfiable. Summarizing, given a set $S$ of clauses, you can use Resolution to derive the empty clause if and only if $S$ is unsatisfiable.

Unfortunately, sometimes generating the empty clause takes an exponential number of steps. Let $G$ be a random 4-regular graph on an odd number of vertices $n$, and let us associate with each of the $2n$ edges a variable. Consider the set of $\Theta(n)$ clauses stating that for every vertex, the XOR of the four variables corresponding to edges touching the vertex is 1. Since the number of vertices is odd, this set of clauses, known as a Tseitin contradiction, is unsatisfiable. Urquhart showed that with high probability, any Resolution refutation of this set of clauses has $2^{\Omega(n)}$ many lines (i.e., clauses); see Jakob Nordström's lecture notes.