Shorted out Uno while successfully running soleniod actuation sketch

No, it might just be in latched-up state (sometimes happens with CMOS if signal voltages higher than power supply) - power down and retry cautiously. If still getting hot something is wrong.

Also there is a possibility of a short-circuit somewhere leading to overheating (and maybe damage) - a multimeter is invaluable in tracking these down.

A good rule when connecting up circuits is "don't apply power until you have double checked the whole circuit". This saves money over time!