All Pis should work. The Raspberry Pi3 model B though is the fastest and runs great on mine. It also ships with the BrickPi3 so there should be no issue.
For question number 2, anything you do on your computer should work. Downloading packages, streaming YouTube, whatever. That answer your question?
For number 3 power banks will work, but should be 5V and 2.5 Amps. Technically you can go with 2, Dexter Ind does it with there Pi power supply, but 2.5 is better. If your interested in using a power bank see this thread.
Dexter Ind also finally released a rechargeable battery pack for the BrickPi! https://www.dexterindustries.com/rechargeable-battery-packs-for-gopigo3/