Assuming they're using a USB charging port, that's 5V at 500mA by modern standards. Of course, that means they would require a full thousand hours to charge according to the numbers on the sticker sheet, so that doesn't quite sound right. Sticking with 5V (Shinkis aren't physically big enough to run wires capable of 100V through them unless they're room-temperature superconductors, which they might very well have in the future, without melting due to heat), and a charging time of 10 hours (seems reasonable for overnight), then either the chargers can supply 50A at 5V via induction, 2A at 125V RMS (approximate wall voltage), or something in between. Rough estimates says that those induction coils should be huge either way; I'm going to start theorizing that either whatever spacial funny business visualizers can pull is always active inside Shinkis, or in the future they've found room-temperature superconductors. That, or Konami could just be pulling things out of their screwholes.
As for the directed-energy weapons, the square-cube law actually works in favor here (as it tends to with Shinkis); the amount of plasma discharge is so small, and the range so short, that they might be plausible with the already-absurd amounts of power Shinkis seem to use.
But, as Fenix said, we're killing catgirls here, and I don't feel like flipping through my textbooks to find induction equations and then try messing around with them, so I'll stop there.