Internship concerns [closed]

You have the right to ask what she will do with the video chats. She also might have the right to tell you that she is recording them without your consent. Sort of a "too bad, so sad, I'm the boss." type of thing. The actual legal details will depend a great deal on where you are. In my state in the U.S., if I am engaging in a voluntary meeting, the other party can audio record me (for non-commercial purposes) without my consent. (The laws are different for an dissociated 3rd party). You may not have a large legal leg to stand on. In most universities in the U.S., you can be recorded without your consent or knowledge if you are engaging in a classroom activity and the professor has initiated the recording. States with one-party or mixed consent laws on recordings almost always will allow universities to record anything considered "classroom activity" if the professor (or even an engaged student) is aware that the recording is taking place.

As for the new syllabus, perhaps she is just trying to formalize what she expects from you. If she wants you to complete additional work, then you can of course address your concerns with her. This is especially true if you have a syllabus for the previous agreements for the internship and the new syllabus does not agree with the old one.

ADDENDUM: Perhaps this is just my cynicism speaking, but we do need to state the fact that the (professor/university admin) to (student) relationship is often very one-sided in terms of power. Short of committing a legal infraction, most professors will be allowed to get away with about whatever they want. They can give you extra homework. They can make you jump through hoops to get a good grade. They can even make you take unfair and extremely difficult tests. School is not "fair."