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Thanks for reaching out!
For the first question, I must mention that the port you mention is the USB on-the-go port of Edison. Therefore it complies with the on-the-go standard (you can learn more about it in https://en.wikipedia.org/wiki/USB_On-The-Go). However, for this specific question you must know that this port works as a standard USB port being powered with 5VDC/500mA. So if you are going to use 11.1VDC, I suggest you to stick with J21.
For the second question, since you are using the mini breakout board, it is not recommended to power an external device from the Edison. Instead, I suggest you to use a 3.3VDC voltage regulator that is powered from the same power source of Edison. This regulator can be powered with the same 11.1VDC that Edison is powered. The only thing you have to consider with this is that you have to join both circuits ground lines together, otherwise both chips won't be able to communicate with each other.
I hope this information helps you.
I am currently powering my Edison with a 3S 11.1v lipo on J21 and it works quite well. Also I am using the Edison's 3.3v output (J20-2) to power an MPU9250, and can communicate with it directly with I2C1 (J17-8 and J18-6) even though those lines are technically only 1.8v.
The newer MPU9250 may be able to run on 1.8 volts, which might be why I can get away with doing this, but it'd would be worth trying the same set up with the MPU6050. If it doesn't work you could always buy the MPU9250 here https://www.amazon.com/Diymall®-Mpu-9250-Nine-axis-Attitude-Acceleration/dp/B00OPNUO9U since it's only $12.50 and forgo the extra voltage regulator circuit.