yes

Luxli Cello will accept 7 to 15 volts DC though its 2.1 mm barrel jack. Its easy to make custom DC cords for whatever kind of mobile power supply you may own. For example, I made three kinds of cords: 1) Male 2.1 mm barrel plugs on both ends to power a single Luxli Cello from an OmniCharge 20 DC in/out port (2.1 mm barrel). 2) MR30 to male 2.1 mm barrel plug to power a single Luxli Cello from an OmniCharge Ultimate 150 watt DC out port (MR30). 3) MR30 to three male 2.1 mm barrel plugs wired in parallel to power three Luxli Cellos simultaneously from a single OmniCharge Ultimate. Bonus: the small AC to DC charging brick that comes with OmniCharge 20 and the larger brick that comes with OmniCharge Ultimate will also power Luxli Cello.

Yes, an NP-F750 battery and charger are included with the light.

Well, if you're mounting the light on a camera, one is all you need. They put out a pretty good amount of light, but two will only work if you're lighting a small area. Say someone's torso

These ARE underpowered for anything more than interior closeups.

I leave the battery inthat way, if anything interrupts the power source, the battery keeps the light on and connected to the app over Bluetooth without interruption. More than once, that feature has saved my bacon on a shoot.

I'm using a Beyer M58 mic case, zippered.

Unfortunately, the Angler Collapsible Softbox for 6x12 LED Lights B&H # ANLCSB1206 is too large to fit the Luxli Cello 10 LED Light nor are there any available softboxes for it at this time.