Transformer construction diagram
I have seen several diagrams like this but this one is confusing.

It doesn't tell you the size of wire, the start and finish do not match the circuit diagram and the number of turns is not explicitly given though you may be able to count them from the drawing.  It also does not explicitly state the core and bobbin.

My best guess would be as follows all winding start and end on the right and side of the bobbin.

Half the primary is wound first start on pin 3 wind 8 turns across the full width of the bobbin secure with a single layer of tape now wind back another 8 turns (16 total) and finish on pin 4.

Next we need basic insulation which typically may be 3 layers of tape.

Next we have the secondary start on pin 12 wind 8 turns across the full width of the bobbin, secure with tape, lay the wire back over the winding and finish on 9.  

Next we need basic insulation which typically may be 3 layers of tape.

Next we have the auxilliary start on pin 6 wind 8 turns across the full width of the bobbin, secure with tape, lay the wire back over the winding and finish on 5.

Next we need basic insulation which typically may be 3 layers of tape.

The other Half the primary is wound first start on pin 4 wind 8 turns across the full width of the bobbin secure with a single layer of tape now wind back another 8 turns (16 total) and finish on pin 3.

Finakkt we need basic insulation which typically may be 3 layers of tape.

Fit the cores and you can test it.  Typically flyback transformers are designed for a specific inductance which may require a controlled air gap depending on type of core.

To check inductance you need an inductance meter to check phasing and turns ratio you need a signal generator and scope or DVM. Using a DVM is more tricky as it won't show phase but by putting windings in series and swapping the phasing of one you will see that the output voltage is higher when both windings have the same phase.

Note: When this was originally written only the final construction drawing was in the question so this answer is based purely on that so the construction may have changed but this should illustrate the general approach.  

I have answered your question about how wiring diagrams typically work and how to check a transformer but I am not convinced the transformer is necessarily the problem.