Is it safe to power the Uno with 18v? So I'm aware the Arduino Uno can take 6-20 volts, with a recommended voltage of 7-12. But I have an unregulated 12v power adapter, which puts out about 18v with no load.

Most of the time the Arduino won't be doing much of anything but powering a 16x2 LCD, which should take only a couple mA.* I have a 10k ohm resistor on the back light, so it should only take 0.5mA.
*I don't have a datasheet, but from what I've read 1-2mA is reasonable

The Arduino won't be doing anything else that would draw any meaningful amount of current (except looping to check for input, etc). Online sources generally indicate the power draw when not doing anything is a little under 50mA, so I'm guessing that's about what the current draw will be. Using the formula in this answer, I calculate the power dissipation by the voltage regulator to be (18v-5v)*0.05A=0.65 Watts.


Is this correct?
Is the voltage regulator likely to overheat?


I'm not really concerned about power usage, because it will be plugged into the wall all the time. Also, the Arduino will be out of sight, but will have plenty of air around it to disipate heat (though not access to a breeze).
It won't do any harm to try it at 18V for a while. Most voltage regulators have thermal protection and just cut out if they get a little too hot. If it works reliably, stay with it.

EDIT: Their Uno page warns against overheating the board, so I would test it for a short while and make sure there is no overheating or strong "plastic" smell coming from the board. If there's enough heat to worry, then it's best not to run like that for extended periods.

If 18V is too high, you could get a power resistor (I'd guess say a 10 watt resistor around 24 ohms) to waste approximately half an amp from the power supply. Connect this to the supply, in parallel with the Arduino.  It's wasteful (unless your room needs heating anyway) but will lower the voltage (that reaches the Arduino) a little.