Is cohomology of groups all about $H^{i}: -2\leq i\leq 2$?

As Serre seems to be fond of saying, life begins at $H^3$.

The first appearance of a $3$-cocycle seems to be in Teichmüller's article Über die sogenannte nichkommutative Galoissche Theorie... in Deutsche Mathematik 5. He was trying to extend Galois Theory to extensions of a commutative field which are not themselves commutative. Teichmüller's formula for the $3$-cocycle helped Eilenberg and MacLane come up with the right definition of group cohomology in

http://www.jstor.org/stable/1968966 (Group Extensions and Homology, Samuel Eilenberg and Saunders MacLane, Annals of Mathematics Second Series, Vol. 43, No. 4 (Oct., 1942), pp. 757--831).

Let me just give one example where $H^3$ is useful. Let $p$ be a prime number, and consider a finite extension $K$ of the field $\mathbf{F}_p((t))$ (or of $\mathbf{Q}_p$, which you seem to know). Let $\bar K$ be a separable algebraic closure of $K$, and $G_K=\mathrm{Gal}(\bar K|K)$. It is an absolutely crucial result that $H^3(G_K,\mathbf{Z})=0$. That's what allows you to lift projective representations $G_K\to\mathrm{PGL}_n(\mathbf{C})$ to linear representations $G_K\to\mathrm{GL}_n(\mathbf{C})$. See for example

https://eudml.org/doc/142305 (André Weil, Exercises dyadiques, Inventiones mathematicae (1974) Volume: 27, pp. 1--22).

Here's a number field example, analogous to Dalawat's local field example. Let $K$ be a number field. One can prove that $H^3(G_{\overline K/K},\overline K^*)=0$. (Although interestingly, there may be finite Galois extensions $L/K$ with $H^3(G_{L/K},L^*)\ne0$.) Anyway, Tate's cohomological construction of the pairing on the Tate-Shafarevich group $III(E/K)$ of an elliptic curve $E/K$ proceeds by first constructing a 2-cochain and then using the fact that $H^3(G_{\overline K/K},\overline K^*)=0$ to find a "correction term" that turns it into a 2-cocycle, thereby giving the desired element of $\text{Br}(K)$. So the whole construction depends on the calculation of an $H^3$.