So in theory if I was driving 1000 LEDs it would be fine as long as the power supply could handle 50A? The ground pin on the Arduino doesn't care how many amps are in the circuit coming from an external power supply?
And without using the breadboard, can I just solder 3 wires together?
That would be a good way of connecting it.
The idea is to have the lights blinking with the music so 5A is the absolute maximum, on average it should draw much less than that.