The black things on the backs of the keys are carbon pucks. They are just silicone with a high carbon content to trigger the capacitive circuits on the board.
Battery thickness. Depends heavily on your requirements. Coin cell batteries can usually be found that are ~3mm thick. You’ll find those in things like the Apple remote controls. Li-ion or Li-po packs like those used in the iPhone or Ipod touch are usually only about 5-6mm thick.
For the induction coil, see if you can find any of those commercial cell phone charger packs. The coil itself is probably very thin, but requires a large amount of surface area.