Electronic – Using a transformer with 400 volt rated primary side with a 120 volt line

1) Winding direction doesn't matter as long as you keep it consistent. If you are adapting a lathe or drill as a winder, the winding direction will be determined by the machine.

2) If the small windings are at approximately the same potential they can share a layer, otherwise you must provide sufficient insulation between them. Sometimes you may have to separate layers to reduce capacitance between them, or interleave them to reduce leakage inductance, but that is too complex to discuss here.

3) Winding coils side by side is simpler and provides better voltage isolation especially when the sections are different bins on the bobbin. That is essential for high voltage supplies and makes mains voltage safety approval easier.

However it increases the leakage inductance, and that matters for high frequency applications like SMPS. The only reasonable way to know if you can do this is to wind one that way and measure the leakage inductance. If it's out of spec (or even close to it) wind another in layers and compare them.

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.