Did Sherlock really have a sister?

I'm not used to seeing the expression "there exists" as opposed to "there is/are", is it a common expression? What is the difference between "there exist" and "there is/are"?

I am unsure about British English, but we don't usually say "there exists" in American English. We usually say, "there is/are." I believe "there exists" here is referencing a specific place where the witch's flowers are located. It's not a common expression in American English, but it may be common in British English.

Is "rub" a noun here?

Yes, the word "rub" is used as a noun in the case of "back rub". The word "rub" is not often used as a noun, but it can serve as a noun in a few cases. Ex: He disinfected the cut with an alcohol rub.

What does "even" mean here?

Hahaha, I love this video! In this clip, "even" is used to emphasize his statement. He is telling the students that they better not try to play any tricks on him or to mess with him. "Even" in this context is typically used to emphasize anger, seriousness, or surprise. Ex: Don't even think about it. Ex: He has never even heard of Jennifer Aniston! Ex: You don't even have a chance of winning the lottery.

What does "have down" mean? Something like "understand"?

To "have [something] down" is similar to "understand!" You're right! It's to be knowledgeable or proficient in something so that you can do it well. It can also mean to have written something down. Ex: I have the presentation down. So I'm ready to present it today. Ex: She's got the whole process of coding down. Ex: I've got it down on my notes here.

What does "dead mark" mean? Is it a bad omen, perhaps?

Not quite! In film terms, a "mark" usually has to do with letting the actor know where they have to stand when filming a scene. I'd say a "dead mark" refers to an exact mark, as "dead" can mean complete or certain. Ex: The road is dead ahead. => straight and exactly ahead Ex: I need a mark to know where to stand.