Well I run in the Adirondacks, but here's my take on those things:
Headlamp - always, even on a morning run. Modern headlamps are so small and light you can carry them without noticing them. And if you do end up out at night, they make all the difference.
Rain gear - depends on season / forecast. During really hot summer, I don't bother, even if it might rain.
Water - depends on length, sometimes carry a filter. A lot of the serious guys just drink the brooks in the interest of time. I'm slow, so I don't need to take that chance.
First aid - not much. After decades in the woods, and as an EMT, I've found that the two categories: "it will get better by itself" and "you can't fix it anyway" cover the vast majority of injuries / illnesses. Cuts, bruises, minor sprains - just keep going. Broken leg - you can't fix it with anything you can carry and you will need evac. The rare exception might be a deep cut where it's hard to control bleeding, or a known illness where you can carry a specific symptomatic med.
Tell someone where I'm going - always. It's free and easy to do, and results in a prompt and less expensive search and evac, rather than a stupid Aaron Ralston situation.
TCD