Electronic – arduino – Peltier with Mosfet – Heat Issues and Alternatives

If your heater is only resistors and you're going to PWM them anyway, then connect them directly to the 12 V supply. The linear regulator will only waste a lot of power and make heat that you have to deal with. Arrange the heater resistors so that the heater gets a little warmer than you need with 12 V applied. You then use PWM to throttle that back.

For the Peltier, you don't want to use pure PWM. That is not only much less efficient, but can also damage the Peltier. In that case, you want to use a switching power supply. That can be as simple as the PWM you show, but with L-C filtering to keep the current thru the Peltier reasonably constant for any fixed PWM duty cycle. The general rule of thumb is to keep Peltier current ripple to 10% or less of the average. The temperature feedback closes the loop, so you don't have to explicitly regulate volage or current thru the Peltier. The control loop adjusts the PWM duty cycle according to the temperature feedback.

There is more information about driving Peltiers from a digital PWM signal here.

No, the circuit does not contain a full bridge.

Think of the TEC being L1 in the image. To make the TEC switch from one side cooling to that same side heating current needs to flow in the opposite direction through the TEC. One direction would be Q1 and Q4 ON and the other two transistors OFF. To switch current directions turn Q2 and Q3 ON and the other two OFF.

Since this circuit only contains one transistor the TEC cannot have its current direction reversed. It looks like your code will just shut off the current altogether.