Is the $G_2$ Lie algebra useful for anything?

I don't know if these rise to the level of "useful," but:

• Yang-Mills theory with gauge group $G_2$ is interesting because $G_2$ has trivial center. So people simulate it on a lattice, try to understand in what sense it might be confining, how string tensions scale, if it has a deconfinement phase transition, and so on. The idea is that looking at a group with no center provides an interesting window into which phenomena in gauge theories rely crucially on the existence of a center and which do not. One recent paper (selected more or less at random from a search; I don't know this literature well enough to make useful suggestions) is here.
• M-theory compactified on seven-dimensional manifolds of $G_2$ holonomy gives rise to four-dimensional theories with ${\cal N} = 1$ supersymmetry. I don't know the earliest references (probably this knowledge goes back to early work on supergravity before M-theory), but one place to look might be this paper of Atiyah and Witten.

Yes.

G2 shows up often, starting with atomic physics (perhaps Racah is the first; see R. E. Behrends, J. Dreitlein, C. Fronsdal, and B. W. Lee, “Simple groups and strong interaction symmetries,” Rev. Mod. Phys. 34, 1 (1962).). You will find some refences in my 1976 Phys rev paper on cns.physics.gatech.edu/GroupTheory/refs . I have whole folder of physics G2 papers, but now I see I did not bother to enter G2 history into www.birdtracks.eu.

Nobody's perfect. Sorry

Predrag (for responses, email to dasgroup [snail] gatech.edu, I sometimes look at those. Pure accident I saw this question...)