Yes, CYUSB3333 (HX3C) is capable of receiving power from USB Type-C instead of DC power supply as in Dongle designs.
1) Yes, you can add a BUCK BOOST Charger and a battery but it needs lot of code change from the existing one.
2) The answer is YES for all (a), (b) and (c).
Thanks for reply, When these IC will be available for Local Purchase.
All these IC are coming soon . :)