Electronic – Not sure how to go from PID error to PWM duty

I know exactly where you're coming from. The mistake you're making is assuming that the integral term drops to zero in steady state. This is not the case, and, indeed, is highly dependent on implementation details.

First off, understand the integral term in a mathematical PID is the integral from the start of time (or, well, the system) and not "error over last few cycles". Your implementation of PID or PI should not cause the older contributors of the integral term to drop in relative weightage of the I term. Let me explain. When writing the I term's code, the first instinct is to assume that the term will diverge, crossing the variable size and overflowing, and people attempt to fix this using moving averages, degrading the weightage of the older values, and all sorts of strange gimmicks. This should not happen in a properly implement PI or PID system. Instead, you should simply calculate I as I = I + Ki*Error.

The baseline level required to maintain the system, which you mention in your question, must be provided by the I term. Since you do not know how much this is apriori, you must allow the controller to discover this value for itself. That, in fact, is the job of the I term. The Ki value should be small enough for the controller to converge before it overflows. Some thought about how this works on paper will help. Try to visualize the process, not specific boundary conditions. One thing that you should keep in mind is that the I term is not constructed from absolute value of error. It includes both positive and negative values of error.

Further, imagine the condition where the controller is just reaching the steady state. You will realize that I is not necessarily zero at this point. Indeed, I is actually the baseline control force you mention in the question. If the state actually remains stable, and if error from here on in is continuously zero (or zero averaged over time), the value of I will remain as it is.

Now, when it comes to real implementations, the problem you will face is that even with a small I, by the time your system reaches the set point, I may well have saturated. The system will then have to err in the opposite direction for a long time to rid itself of the I term it accumulated while it was still reaching the set point. In fact, I've noticed that PI and PID work best for a single set point, and degrade when you have to keep changing that point by a large value. A big contributor to this is the fact that I has high inertia. Tuning the value of I is possible to keep rhe controller functional, but when the system itself responds to stimulus slowly (say you're heating a block of metal), tuning is often difficult. Instead, what can help is to activate I only when. The system is within a certain threshold of the set point. When you change the set point by something greater than this threshold, clear I and disable it (use only P/PD control) until you reach close to the new setpoint. By doing this, you add another tunable parameter (the threshold), but it makes setting both Ki and the threshold easier than setting Ki by itself to be optimal for both situations.

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.