Electronic – Amplify PWM signal from 0-5V to 0-10V

Q1: The controller will find the PWM value to keep the temperature at the setpoint if the PID is reasonably well tuned. If it not tuned well, it may oscillate around the setpoint or be very sluggish in approaching the setpoint, and also slow to respond to demand changes (such as a change in ambient temperature).

Q2: Yes, if you make an educated guess as to the values it is possible to get stable control with a try or two. Or use a self-tuning algorithm. And/or a manual method such as Ziegler-Nichols.

Q3: They will work reasonably well for some range where things stay more-or-less linear. If you triple the wattage of the Peltier, for example, you'll probably get a lot of overshoot. In most applications (not all) the tuning parameters are fixed numbers for a given range of setpoints (perhaps all valid setpoints).

It's not always possible to get the control you need with a single control loop, sometimes nested loops are required. I've gotten control down to the \$\mu K/\sqrt {Hz}\$ level but it's not easy. Your 0.05°C may be difficult or not so bad, depending on what the system looks like.

I haven't tested this, and its off the top of my head, but based on http://www.stuffandymakes.com/blog/2012/07/08/disney-glow-with-the-show-ears-teardown the LEDs are common anode with the microcontroller sinking current (open drain). Using a optocoupler on each line, with the led amp providing the power switching, you should get exactly what you want. All you need is the optocoupler in between vcc and the rgb led (one per color).

^{simulate this circuit – Schematic created using CircuitLab}