Why doesn't Epsom salt dissolved in water act like sulfuric acid?
Well, as you might have guessed, it's the $\ce{H+}$ in sulfuric that is dangerous, not the $\ce{SO4^2-}$. So, Epsom salts, being $\ce{MgSO4}$, aren't very dangerous.

In the same vein, table salt ($\ce{NaCl}$) is perfectly OK despite hydrochloric acid ($\ce{HCl}$) being rather nasty.

And $\ce{H+}$ is only corrosive in high concentrations, mind you - otherwise we'd be burning ourselves drinking orange juice, which itself has a small concentration of $\ce{H+}$. That also answers why water doesn't kill you: only a tiny fraction of it (roughly 1 in ten million molecules) actually undergoes autodissociation at any one time. It's not about the actual number of $\ce{H+}$ ions, but rather the concentration. If you dilute $\ce{HCl}$ enough you can pretty much just drink it. (Please don't be stupid and take this as a licence to do a home experiment and drink $\ce{HCl}$.)