Temperature stabilization with PWM , PID controller with delay system
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.