How does gamma trading depend on $K$?

If we think realized vol > implied vol, then we might go ahead and delta hedge a call, hoping that profits from gamma outweigh the decay.

Question: What should $K$ be on the call? ATM? If so, why? What happens if its OTM/ITM?

There is much more to exploit when gamma is highest. Your daily gamma PnL is $0.5\Gamma(X_t-X_{t-1})^2$, so you would probably prefer to have $\Gamma$ highest which is near the money. To be precise, $\Gamma$ is highest when $d1=0$ which occurs for $K=Fe^{0.5\sigma^2t}$. Of course, I am assuming a European option and that the Black Scholes model is correct of course - either way, close to ATM is what you want.