Two Power Supplies Is Okay
You can drive the Arduino Uno from the external power source and have it plugged in your computer with the USB cable at the same time. Having a recently purchased computer I was worried that having an external power source could feed back into the USB port and somehow overload the computer. This Stack Exchange answer goes over the details and the schematic about why this is possible.
A 9 volt battery can drive the Arduino and a Pixel Strip
Working on a halloween costume, I was testing running the Arduino and a strip of LED pixels from a single 9 volt battery. I works, but I don’t know for how long…
The Vin Pin = The Power Input
The Vin pin is used to drive external boards and accessories. It sports the same voltage as the external power input. In my case I drive for the Plexi Project, I drive the Arduino with a 12v power adaptor and I can drive my LEDs from the Vin. No need to splice the input from the power adaptor separately.