Hi friend, you do not hate winding transformer, I am sorry I kike, it is very interested if you deep study, I reply your question.
(a) ferrite cores , becasue power is less than 2.4W and 200K frequency , so you only need RM4 if you like RM configuration,EF13 or T14 are also good.
(b) number of turns on primary (and therefore secondary) , according to formula: V=L*di/dt, V is minimum DC voltage, I is Irms, t is time of flow, you have frequency and duty cycle, you can know t value.
(c) wire gauges (only rough ideas using Ohm's law) for power transformer, current density is 3-5mm/A, you know Irms and current density, you can know wire diameter,which type will you select depend on your design standard,EN61558 or EN60065 or EN60950 or other.
(d) size of core to be able to transfer the energy required , please check (a)
(e) primary or secondary wound first (innermost) , not special requirement, however you should be consider low leak inductance and capacitance
(f) direction of windings, all winding should be keep same direction normally
Addition, will transformer applied either flyback and forward, if it is flyback transformer, it must open air gap.
If you have other question, I am glad to help you.