s it safe to run the arduino off of the same 15v power supply
I would say yes. The major limiting factor is the voltage rating of the electrolytic capacitors on the input. On my Uno these are 25V so 15V should pose no problems.
Yes the regulator will run hotter but this is unlikely to be a problem unless you draw a lot of current off the 5V lines. For a discussion of how much hotter see:-http://www.thebox.myzen.co.uk/Tutorial/Power.html
and the specific issue of the arduino board is shown here:-http://www.thebox.myzen.co.uk/Tutorial/Power_Examples.html