If you want to understand the quantization of gauge theories and the BRST procedure in particular in detail, the best reference is probably *"Quantization of Gauge Systems"* (QoGS) by Henneaux and Teitelboim. Requirements for a BRST symmetry to exist is that we have a gauge theory - or equivalently a constrained Hamiltonian theory - and that this gauge theory fulfills regularity constrains I discuss in this answer of mine (but which is also taken from QoGS).

There is no notion of multiple "BRST parameters". The BRST symmetry generator $\Omega$ is the generator of the BRST differential $s$ such that $sF = \{F,\Omega\}$ where $\{-,-\}$ is the Poisson bracket on extended phase space (i.e. including ghosts, and taking into account the grading w.r.t bosonic and fermionic variables). I explain the construction of one part of it in this answer of mine and its rough connection to the Lie algebra in a generic Yang-Mills theory in this answer of mine

The proof that the BRST differential exists for a general gauge theory obeying certain regularity constraints and has a generator is rather technical. It requires proving the "main theorem of homological perturbation theory" (theorem 8.3 in QoGS) and then using the differential $\delta$ on the ghosts discussed in my answers linked above. Once you have the generator $\Omega$, the infinitesimal BRST symmetry is just $F\mapsto F + \epsilon \{F,\Omega\}$, with a single parameter $\epsilon$. The number of ghosts depends on the number of constraints in the Hamiltonian formulation of the gauge theory, and the BRST operator in the simplest cases is given by $\eta^a G_a$, where the $\eta^a$ are the ghosts associated with the first-class constraints $G_a$.

Note that the BRST symmetry acts on a very different space than the gauge symmetry - the BRST symmetry acts on the extended phase space with ghosts (and ghosts-of-ghosts, etc.) and only BRST-invariant quantites are physically meaningful, while the original gauge symmetry (with potentially multiple parameters) acts on the original phase space. It is subtle and dangerous to try to compare their actions directly.

This post imported from StackExchange Physics at 2020-12-04 11:34 (UTC), posted by SE-user ACuriousMind