Parts that are more critical typically have some form of nondestructive testing to validate the quality of the part.
The answer is "yes" to all of your questions, followed with "depending on the details"
Yes, a 'shut down' can be issued if certain parameters are exceeded on any particular sensor or error code. That's pretty commonplace and standard. You can watch the monitor for FDM machines, for example, to ensure something doesn't go haywire - part shifts, nozzle blockage, or your support structure was inadequate and the part is drooping, etc. There isn't -much- you can do (within practical means) for those situations, as the machine is going to blindly keep-on-truckin' since it senses no problems.
With experience, vetted processes, and proper machine care and maintenance, you don't need to monitor things AS MUCH.
The requirements and solutions will vary based on machine type and typical part run time, I expect.