Can't figure out how to distribute the power
The best option is to connect the 5V/2A output of the UBEC to the 5V pin on the Arduino. In my experience, 7.4V is not enough to feed into the input jack on an Arduino especially if it is from a battery, as the voltage will drop when current is being drawn by other loads. I have built an Arduino controlled robot using a 6V NiMH battery and brushed DC gearmotors (so no ESCs or UBEC) and used a $5 5V 1A step-up/step-down converter so that even if the battery voltage dropped below 5V, it could still power the board. However, since you have a UBEC that is not in use, use it. Note that if you put voltage into the 5V pin, disconnect it when you are connecting the arduino over USB or you might damage your board. 

As for inductive charging, I don't know of any off-the-shelf solution designed for charging a 2S lithium ion battery. The best option for charging would probably be an easy way to open the ball to access the batteries.