How do I feed NRF24L01 from Arduino Pro Mini?
can I feed NRF24 directly from Vcc pin?


NO: the pro mini is either 5V or not strong enough to power the RF24.
EDIT: the 3V3 pro mini "could" power the RF24 via VCC, if you use the cap and if it's not the PA&amp;LNA model (or in full power mode). Best practise is to use an external regulator.


  if I fed NRF24 directly from battery and reduce 3.7V to 3.3V via
  another TIP122 transistor?


NO: lipos are 4.2V when fully charged, and they can be used until 3V (I mean 3V sag voltage, when the load is powered, not when disconnected): External 3V3 regulators is the way to go.


  Do you also think it is a reliable way of controlling the motors'
  speed?


These are PWM output, so yes, why not? Also, add flyback/freewheeling diodes (and -maybe- 100nF polyester caps should help) between the motors terminals.