Suppose I am a Professor working for a reputed university. Am I allowed (by the university) to make teaching videos of my courses at home and publish them on a platform like YouTube or somewhere else and monetize them?
Standards and laws differ (widely) from place to place. I never worked at a place that constrained me from such things, but some questions on this site imply that in some jurisdictions your entire intellectual output is claimed by your employer even when it is a university.
But you can learn the specifics of your case by speaking with some legal officer of your university. They can point out what contractural and/or legal rules apply to your specific case.
As an example, though not the same thing, some universities claim patent rights on any patentable material produced by faculty and may (probably will) share in any proceeds that accrue. In such cases patents are filed in the name of the university. While this seems like a bad thing it also means that the university assumes the cost of filing the patent (lawyer fees...) and defending it. These costs can be substantial.
Note that the fact that you produce these "at home" is probably not relevant if you have contractual obligations.
It depends on the wording of your contract with the university.
In general, professors monetize their academic work all the time. Be it through writing textbooks, consulting, patenting, building startups based on their research etc.
Universities may be entitled to royalties from any such activity, though the exact amount may vary. For example, Stanford received $114M from royalties in 2020 alone (it's worth mentioning that Google pays them royalties).
If you are planning on making online lectures and monetizing it, the best course of action is probably to let the university know that you're planning on doing it. They'll help establish who holds intellectual property rights to the material exactly.
Most institutions have a legal team whose job is (among other things) to protect the university's proprietary rights, and they'll know what to do in this situation. That said, they are protecting the university's interests, not yours. If you are serious about this (and not just uploading lectures for a few extra bucks off of YouTube ads), then it may be worthwhile to consult a lawyer who specializes in IP/labor laws.
This cannot be answered because there is no general agreement between universities and teachers. Whether or not you are allowed to do this simply depends on whether your university allows this or not.
For your information: at some universities this is definitely allowed, but there will usually be explicit agreements about who owns the IP and who can profit from such videos.