I was thinking, we are taught to perform code reviews, inspections etc. but in reality, does anyone really do formal (Fagan) inspections? I cannot find anything on the topic.
Fagan inspections are meant to catch errors in early phases of an explicitly waterfall process so that the team is less likely to have to move back up the process to fix errors, because doing so is extraordinarily expensive.
Waterfall processes in general and the work products that support them have fallen very much out of favor for consumer software because the cost of being late to market is much higher than even the cost of rework in most cases.
However, if you are writing something that must work correctly the first time, then a waterfall process and Fagan inspections are the way to go. I expect they are still extensively practiced within organizations like NASA and aeronautics. I once chatted with a programmer from Honeywell and he commented that he spent so much time in inspections that he was lucky to write one function a day, but that one function was so well specified and reviewed that it was virtually impossible for it to be incorrect. (Note the "virtually" in the sentence. The Mars Polar Lander and Ariane 5 maiden flight are famous examples where even that level of inspection failed.)
Yes they do, I would say that is what is usually known as unit testing - not code reviewing.
From the linked Wikipedia article: "Fagan Inspection defines a process as a certain activity with a pre-specified entry and exit criteria. In every activity or operation for which entry and exit criteria are specified Fagan Inspections can be used to validate if the output of the process complies with the exit criteria specified for the process."
Unit tests test units of code by pre-specified entry and exit criteria. Note that no where in that article it says that the 'review' has to be performed by a human reading code with their eyes. It is explicitely about comparing input and output, and says nothing about how well-written a piece of code is. Unit testing is an automated way of reviewing if a piece of code with a pre-specified entry, fulfills the pre-specified exit criteria.
So yes, many software developers do Fagan inspections - but those are not code reviews, they are unit tests.