Electrical – Transformer voltage step-up and heat

You should consider separating the two cores into two transformers like this: -

^{simulate this circuit – Schematic created using CircuitLab}

R1 and the two cores looks complicated the way you have it drawn but the only thing linking the two cores is the current flowing through R1 - that's how I read the diagram and that's what I've redrawn.

Because these are 1:1 perfect transformers with no leakage, you can then move R2 from the right of XFMR2 to its primary and discard XFMR2. Now you have a 9 ohm + 6 ohm load (15 ohm) on the secondary of XFMR1 and, by the same technique this load can be assumed to be directly connected to the input supply voltage of 12V.

This gives a power of: -

P = \$\dfrac{12^2}{15}\$ = 9.6 watts

EDIT to include rationalized picture and calculation of magnetization current: -

Although not shown, the 6 ohm load resistor is connected to the final secondary winding and the total referred load is therefore 15 ohms causing a current of 0.8A. This is load current. Primary magnetization current is 0.6A because \$\sqrt{0.8^2 + 0.6^2}\$ = 1A.

A primary mag current of 0.6A implies the primary mag inductive reactance is \$\frac{12}{0.6}=20\Omega\$. Therefore inductance is \$\frac{20}{2\pi\cdot 50} = 0.0637 H\$. From this you can calculate flux.

Yes, a transformer will worth the opposite when connected backwards.

You have a roughly 16:1 transformer - for every 16 volts you put in you get 1 volt out.

In a perfect world, for every 1 volt you then put in the secondary you'd get 16 volts out of the primary. So 5V p-p would ideally give 80v p-p out.

However, it's not quite that simple.

Transformers are designed to work most efficiently at certain frequencies. To get the most out of your transformer you should run at 50Hz, not 100kHz.

Also you ask if it will be a square wave out. No, it won't. It'll be a wave at the same frequency as the input, but it will be far from square. It would look more like this:

You get big peaks at each edge as the magnetic flux changes, but while it's stationary (during the flat bits of the square wave) the induced current in the transformer quickly decays. To get a faithful representation of your input waveform you would have to limit yourself to a constantly changing wave, like a sine wave, or triangle tooth wave. A saw tooth wave would work, but the rapid flux change at the vertical edges of the teeth would probably get a bit spiky.