Electronic – How do European to American outlet change frequency

Flyback topology is one of the common schemes for SMPS of less than a few hundred watts because of the simplicity. These equations give the first order calculation of some parameters you ask for: $$ I_{peak} = \frac{V_{in} t_{on}}{L} $$ Energy stored by the transformer per period: $$ E_{L} = \frac{1}{2}LI_{peak}^{2}$$ Power transferred: $$ P = \frac{E_{L}}{t_{period}} $$
t_period would 10us for 100KHz. t_on is the switch on time, assuming 50% duty as a starting point, it would be 5us.

1. The peak current must be less than the saturation current while accounting for input voltage variation and tolerances plus safety margin. The peak current does not translate directly to output current through the turn ratio.
2. When wired in parallel, the output windings should be capable of giving 9A.
3. When wired in series, the output windings should be capable of 3 times the voltage @ 3A.
4. For SMPS and flyback SMPS in particular, the turn ratio does not translate to actual input voltage vs output voltage.

I don't know how to say much more within a page or two. Possible resources would be some manufacturers' sites such as TI's. There are usually loads of application information. Pick SMPS ICs that seem to meet your requirements and some of them would have online calculators of recommended circuit arrangements.

Transformers generally fall into two types: power and signal. Can any type of transformer be used so long as it's frequency response is what you want? You won't find power transformers operating at 6KHz.

Transformers contain two (or more) coils of wire wrapped around a former. Those coils are inductors and an inductor's impedance increases with frequency. When a former is used, or something to couple the magnetic field of both coils together, iron (or steel) laminated former, ferrite core, then the use of a core will change the relative permeability and can result in a) a stronger magnetic field produced by the inductor, and b) increase in the inductance. Increasing the inductance reduces the frequency response.

Inductance is a function of the relative magnetic permeability of the material in the core (no core, where there is only air is the lowest value) and is proportional to the cross sectional area of the solenoid. The bigger the cross sectional area of the solenoid (that is, the greater the diameter), the bigger the inductance will be and the worse its upper frequency response.

Power transformers are typically designed to cope with 50 or 60Hz and designed to handle large currents.

You'd need to think about a pulse transformer which is designed with a much higher frequency response. You could do calculations when you know the inductance of the coils in the transformer or the manufacturer might give you a graph of frequency response.