"Do people just remember all the rules and go through the formal manipulations of the cohomology groups of class field theory mechanically, or are people actually "feeling" what is going on here. If it is the latter"

Let me be one to, if not advocate, at least defend the formal manipulation point of view. Perhaps ironically, given this, this will be an extremely hand-wavy response: Cohomology problems emanate from failure of exactness of a functor (i.e., "something you want to do to something else"), e.g., $A\rightarrow A^G$ for a $G$-module $A$. So you start with a short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$, apply the "fixed by G" functor, and lo and behold, instead of a 0 on the right, you need a $H^1(G,A)$. Forgetting completely about what this object means/represents/is, it is the thing that is standing in the way of exactness. Then you go and notice that Hilbert's Theorem 90, or some other amazing result, tells you that the $H^1$ that you just ran into is zero. Voila! Exactness.

Now. Sometimes you find that $H^1$'s aren't always zero -- sad but true. So you start studying $H^1$'s in their own right. Some times these are manageable, or have been previously calculated, and some times not. Then you find, by looking at the long exact sequences in cohomology, that you could figure out an $H^1$ you need to know by looking at an $H^2$ (maybe you want an $H^1(G,C)$ given an $H^1(G,B)$ and an $H^2(G,A)$). And then lo and behold -- $H^2(G,A)$ happens to be a Brauer group or something else well-studied. Knowing this $H^2(G,A)$ trickles down to give you newfound knowledge of $H^1(G,C)$, which in turn gives you information about some failure of exactness on $H^0$'s (say, via some new short exact sequence $0\rightarrow C\rightarrow D\rightarrow E\rightarrow 0$ for which one would want $H^1(G,C)$), which is what you were trying to understand in the first place ("The house that Jack built" comes to mind). All this not having any idea what $H^2$ is!

And the process doesn't stop there -- $H^3$'s help you control $H^2$, which in turn control $H^1$'s, etc. I've never run into an $H^4$ in the wild, but they're not that scary for exactly this reason. They're just the thing standing in the way of an $H^3$ computation, and fit into the same exact sequences everything else does -- one leg at a time.

In any case, my point is not that you shouldn't try to understand cohomology groups on an intuitive level. It's that you shouldn't wait for a complete understanding of cohomology groups before you play around with the theorems to see what these groups are good for. The two should be learned in tandem, *maybe* even with a preference going toward being able to use them over being able to intuitively understand them.