Amplify a 12 MHz sine wave

You need to determine how much power you need to transfer. From your wiki, I think you need plenty of peak power, and this circuit won't do that.

You don't actually need a transformer -- a flyback converter (boost DC/DC converter) will work, although you will need a high duty cycle to get 230 V from 12 V at high power. At low power, you can use discontinuous conduction, and simpler control (maybe even a 555).

To get started, use a single inductor between supply (12V) and a switch transistor (rated at > 230 V). drive the transistor on with the 555 for a time so that Ipeak = Vsupply*T_on/L, and Ipeak reaches the peak you need (related to the power you need). Then turn off the transistor, and the collector voltage will spike up (because of the inductance) to a high value -- use your D1 & C7 to rectify that. If you limit may duty cycle from the 555 to < 90 %, your circuit won't run away. Regulated the output by masking pulses when VOUT > 230 V.

You can use a mains transformer at lower frequency, but you have to lower the input voltage to keep it below saturation.

At normal operating frequency the inductive reactance of the primary winding is sufficient to prevent the current from rising too high. At lower frequency the current has more time to rise so it goes higher and saturates the core, reducing inductance and causing the current to spike.

Reducing the voltage also reduces the rate of current rise, and so compensates for the lower reactance at lower frequency. Your 120V transformer should be OK with the primary winding powered by 20V at 10Hz, because that is 1/6th the frequency (10Hz vs 60Hz) but also 1/6th the voltage (20V vs 120V). Since it has a 10:1 ratio (120V in = 12V out) it will perform the other function that you want, 'step down by a factor of 10'.

However it will still only be able to handle the same current, which at 20V is 1/6th the power ie. 8.3VA instead of 50VA. To handle 2A you would need a mains transformer designed for 240VA (120V * 2A) which will be physically much larger. This size increase would also apply to a transformer that was designed to run at 10Hz (if such a thing existed).

To step up the voltage with a mains transformer the input winding must still be designed for 120V (at 60Hz). So what you want for stepping up the voltage by 10 times is a step up mains transformer with 120V primary and 1200V secondary, again rated at 240VA if you want it to handle 2A.

Edit to add:-

NOTE: Although a 240VA 60Hz transformer would be able to handle the power, it probably wouldn't do it efficiently. The power loss at 40W/10Hz would be similar to the loss at 240W/60Hz (perhaps a bit less due to lower magnetic losses at the lower frequency). If the transformer was 90% efficient at 240W/60Hz the loss would be 24W. That same loss at 40W/10Hz equates to just 40% efficiency. You might get the expected maximum amps, but at less than half the expected voltage.

To handle 40W/10Hz efficiently you would need an even larger transformer (or at least one with exceptionally low winding resistances for its size).