I thinks its a transfo (TR1 ) , the voltage on the right side will be 10 times of the voltage on the left side.I don't know the purpose of the C , a filter?
I'd say its a transformer with a ratio of wraps being 1:10. Meaning the side to the transistor is say 10 wraps, and the other side is 100, or something. As long as its output is 10x the input I'd think its fine. But I'll definitely yield to those that know more than I.
This is probably a transformer with a ratio of wraps 1 to 10 between input and output coils respectively, the C3 is probably to cancel any reactance resulting from the secondery coil.
Yes, it looks like part of a flyback circuit. But I would expect to see a diode in series between the transformer (coupled inductor) output and the capacitor, since flybacks are normally used to generate DC power.