To answer the question about the iron core, it is simple to concentrate the electromagnetic flux within the core for more efficient reception of the second coil.

I am not sure but, if we do want to use the same transformer principles to excite the secondary coil installed in some electronic device without the need for a medium, shouldn't the change in EM field be at a sufficiently high frequency? Would keeping it at radio wave frequencies work?

But transferring information and transferring useful power are slightly different things right? Information can be digital signals (transferred using EM waves) and only require very small amount of energy. I'm only a mechanical engineer but I'm guessing the receiving end of the Touch n Go thing has some sort of amplifier to boots input signal from the card. Low frequencies should suffice for this application.

But for powering devices, wouldn't we need a higher frequency for greater current?