Electronic – PID controller best when no model of process available

"C" and "P" aren't constants or fixed gains - but functions of time (or by Laplace transf. -> on complex "s"). In other words, are dynamical systems - where the output depends not only on actual input, but also on its past values. To be short: A input change not reflects immediately on output (instead, need to pass through "C" and "P" blocks) in order to be sensed and compared. This "lag" is reflected by the poles and zeros of transfer functions. The controller "C" is created by design, but the plant "P" does not depend on you - it has its own features. If you want Y = R all the time, then you do not need a control system.

Do you connect the PID such that the controller sets the current to a certain level, or do you connect the PID such that the controller adds/subtracts current to the instantaneous current to reach the desired level?

There are different PID algorithms that do either of those two things. The latter type is called a velocity algorithm (because it controls the velocity of a hypothetical mechanical valve motion rather than the position of the valve). Since the valve performs the integration there were once some advantages to this method- for one thing it automatically stops integrating when the actuator hits the limit, which helps with overshoot due to integrator windup. However, if you are using derivative control it must be double differentiated- which tends to make that contribution very noisy.

By the way, controlling the current through the resistor is usually not a good idea. What you want to do (typically) is to control the power. The resulting nonlinearity from voltage control will usually make the controller hard to tune well over a range of setpoints (it will either be too sluggish or not stable enough).