No, they vary considerably. The power required to drive an earphone is very small, but some FM receiver circuits can consume quite a bit of power, depending on the type of circuit used.
Stereo FM reception often produces unsatisfactory performance whereas mono reception is much more reliable. Since we are anticipating that the major use of this radio will be with a single earphone we are concentrating on achieving best (mono) reception.
AAA batteries are less common and AA batteries usually represent better value for money (in terms of energy per unit of currency). C batteries are too big. PP3 batteries deliver too much voltage for this application (9Volts) and the cost of energy in this format is very high.
No, that type of radio consumes more power. Also, we feel that users like to be able to spin a tuning dial and hear quickly which stations are available.