Why is the regulator very hot?
If you power the Arduino with an external power supply the on board regulator has to dissipate the voltage difference between input and 5VDC at the total current you draw from the board. 

As Klaus Warzecha wrote in an answer to another question 


  The Arduino is just the brain, not the power station.


If the output voltage of your power supply is 12V, then the regulator will be dissipating 12-5V = 7 V at the total current drawn. Total dissipated power for eg. 150mA (just an estimate) would be P = V × I = 7 × 0.150 = 1W
The regulator will be pretty hot from dissipating 1W. 

Mind you that the current is not limited to 150mA, enough LEDs can easily increase current and power. If the regulator is stressed (often) enough, sooner or later it will fail. It is better practice to power any load directly from the external power supply, taking the higher voltage into account.