How to run electricity through metal wire to (safely) act as a stylus?
You don't need to run electricity through a wire to act as a stylus. Capacitive touch screens work by sensing changes in capacitance on a screen. Just plopping a coin on a screen won't register as a touch since that coin is an open circuit and can't modify the charge on that area of the screen. The charge on the screen is still the same. Conductivity alone can't hold a charge.

Connecting a wire to the device's chassis should close that circuit and allow the wire to modify that specific screen area's charge. 


Any material is fine, as long as it can allow charges to be attracted/repelled (form a significant capacitor plate). So, a wire connected somewhere (ground hopefully), or a big enough mass should do it.
0 amps should do it. We're talking about static charge.
Thickness is a large factor. A wire's tip connected somewhere does modify capacitance, but not enough for the screen to register. Instead of changing wire size, consider having a conductive end effector (e.g. ball of foil). Adjust until the touches are solid and reliable. 


(said end effector should hopefully not scratch the screen)