I see. I've never heard that term (except as saying when a business is open). Where does it originate in this context?

In US universities, where professors have to balance teaching and research, they hold "office hours" - hours during the day in which they're in their office, with the door open, and students can come in to ask questions, request help, etc.

Ah, thanks! I've never been, so it was an unfamiliar concept.

The context I'm familiar with is university professor/tutor office hours, being the hours you can come by outside of class to ask questions about your coursework. That also seems to be the most similar to how it's used here.

Professors often hold office hours so students can stop by their desk and ask questions or have discussions outside of th class.

Not sure. Most likely in the context of professional consultation.

No, it's a reference to college/university where the professor is available at certain scheduled times (e.g. Tuesday from 12-2 and MWF from 10:30-11:30), and students are free to drop by without an appointment to ask questions about assignments, material, etc.