Yes, in everyday usage it's fairly common to hear "between" used with more than two things. In your example, you can picture it as though the "in between" location is at the center of a triangle, with A, B, and C as the points of the triangle. In some sense, it is as if you are "between" all of these things at the same time.

It's possible, but the context has to be there. "a", "b" and "c" have to be very well established in the conversation. The person you are speaking to will need to know from the preceding conversation exactly what you mean so there is no doubt. For instance, you would not use it if a,b and c were only mentioned once. They would have to have been discussed or mentioned so there is zero doubt in the listener's mind.

The 'between' here is fine. What you need to pay attention to is "parallel" construction. What does that mean? In your sentence, you have "being grounded" (verb/gerund), then 'your sister' (noun) and "your internship' (noun) They should all be the same type of word: between your punishment, your sister and your internship OR Between being grounded, having to deal with your sister, and working at your internship Another example: I like to listen to music, reading, and ice cream. (wrong) I like to listen to music, to read, and to eat ice cream. This is a common error that appears in grammar sections of exams ( where you have to find the mistakes in the sentences.)

Yes you can. While between is usually used to indicate a physical space or time interval separating two points, it can also be used when describing a relationship involving more than two. The cake was divided between the five guests. Between studying, working and fixing up the old house, he didn't have a spare moment to himself.