Two things,
As others have noted, the Arduino cannot drive 100 LEDs without the help of external components. Well, not without likely exceeding the limits of the Arduino. Ditto for the voltage regulator, it's not meant for large loads. That said, Neopixels/WS2812 units don't require a lot of power on command/control pins, just on the power pins. So, you should be able to connect 100 these LEDs to the Mega without issues, but only as long as they do not derive their power from the mega board. At the very least, make it a separate connection to the common power supply.
Furthermore, I would suggest having a deeper look at all the excellent resources that Adafruit has published re: using the WS2812 or NeoPixel as they like to call it. In particular, look into how they want you to hook up the LEDs.
If memory serves, using two power supplies is encouraged, i.e. one for the Arduino, one for the LEDs, with a common ground. That way, if the LED power supply goes down due to overheating or the power quality suffers due to transients, it won't be as likely to affect your Mega. Given the cost of a Mega, I'd also investigate the use of a buffer chip to make it that much harder to smoke the Mega. PJRC.com has some good tutorials in this regard since their MCU runs on only 3.3V and therefore pretty much needs a buffer chip.
I would also verify the output voltage of your external power supply, just to be sure that you don't release the magic smoke from the NeoPixels. Lastly, verify the input voltage range for the voltage regulator / barrel jack on the Mega. IIRC, it prefers something like 9V, not 5V. At 5V input, the regulator should be going into dropout, doesn't seem like a good idea.