To me it's pretty clear what's going on with the double coils. It's a center-tap transformer. Using half-wave rectification is just generally ugly. Let's not go there. Using a full bridge rectifier requires four diodes, i.e. 2*2 diodes in the current path, with their associated voltage drops. A center tap transformer only requires 2*1 diode in the current path, so you shave off a few tens of a volt from the transferred voltage.
As for current limiting, this might very well be done on the primary side, which would also (hopefully) prevent damage to say your mobile phone if you held it up against the coil. This is probably also why this pulsing charge strategy is in place, to let the microcontroller's capacitor recharge every now and then. The issue would not just be low supply voltage, but preventing ripple on the microcontroller rail, which it is implied if the cap inside diode is charged higher than the peak ripple voltage outside the diode at all times.