Don't solder while wearing shorts.
The software PWM seems like such a nice straightforward approach (especially if you don't need the ULN2003s and just directly connect regular 20mA leds with limiting resistors to the Arduino Mega pins).
So if that is basically correct, my set-up with 7 ULN2003's should not overtax the Arduino mega. Right?
Switching that much current is always tricky and I would add some decoupling to each 2003 chip to stop things oscillating, 0.1uF should do for a start, see:- http://www.thebox.myzen.co.uk/Tutorial/De-coupling.html