During surgical procedures or when subduing a patient, anesthesia is often used. Does this put a person into a deep healthful stage of rest like rem stage 5, or is something more like stage 2 or 3? Or perhaps it simply enables the sleep process, and there is no particular stage of REM. However, I'm doubting the latter because doctors are able to measure relatively precisely how much anesthesia to give a patient.
Sleep and general anaesthesia share a few similarities, but also have differences. From what I have read, they are sufficiently different that comparing anaesthesia to a certain sleep phase doesn't make sense. I recommend reading General Anesthesia and Altered States of Arousal: A Systems Neuroscience Analysis, which I found trying to answer a different question I had myself. Here are a few quotes related to your questions, but it's really all quite interesting.
The EEG patterns and other features of general anesthesia generally differ from those of sleep
These patterns are shown in figure 1.
There is similarity between the EEG patterns seen in slow-wave sleep and those seen in phase 2 of the maintenance period of general anesthesia
Phase 2 and phase 3 are the phases the actual surgery is performed in. For example, a difference between anaesthesia and sleep is muscle tone:
in contrast to the drug-induced atonia described above, rigidity and spasticity are typically seen in patients who are in a coma or a vegetative state, and muscle tone is preserved during slow-wave sleep
Generally it appears that anaesthesia is closer to being in a coma than to being asleep.
confusion arises because anesthesiologists use the term sleep as a nonthreatening description of general anesthesia when speaking with patients. A level of general anesthesia appropriate for surgery is not sleep but rather a coma. However, like sleep, general anesthesia is reversible and can allow dreaming
General Anesthesia and Altered States of Arousal: A Systems Neuroscience Analysis (I recommend the Introduction and Implications sections)
Anaesthesia is not necessarily restful, they can cause a symptom called "rebound REM sleep" where patients need more sleep following anaesthesia because anaesthesia isn't like sleeping:
REM sleep rebound after exposure to volatile anesthetics suggests that these volatile anesthetics do not fully substitute for natural sleep
Rapid eye movement sleep debt accrues in mice exposed to volatile anesthetics