By checking the Mini-Breakout board schematic, I don't see any issue in powering the board with a power source of 5V/1A. However, make sure it provides 5V. If it provides a different value, the Mini-Breakout board and the Edison module could be damaged permanently.
On the other hand, I haven't had issues using WiFi or Bluetooth while powering the board through the USB port connected to my Laptop (I use a docking station, I must say), so you shouldn't have issues using the Power Bank. Make sure the USB port provides a stable output; most of the times those kinds of issues are due to unstable USB outputs.
Finally, regardless of the power source you use, I don't recommend you to connect the LEDs directly to the Edison's IO. I suggest you to use the Edison's IO to activate a transistor, while the transistor is the one that turns ON/OFF the LEDs. The Edison's IO provides 1.8V/3mA, connecting LEDs directly to them might damage the module.