1 He played a part on the music video. Is the sentence correct?
He played a part IN the music video.
2 I called my parents home. Meaning you called your parents when they are home. Right?
You could say, "I called my parents while they were at home." You could say, "I called my parents at home," but I don't think that's very natural.
3 What is the term for the office for suject teachers at school? a subject teacher office.
The [subject] teacher's office, maybe. I'm not sure exactly what you're asking. You could say, "The English teacher's office."
4 After you drink up coffee, there is something left at the bottom of the coffee cup. Do you call it, “coffee dreg,” “coffee scum,” or “ coffee residue?”
People might call it the dregs, or the coffee dregs. Not scum, usually.
5 pass/ tide ovr the crisis / financial difficulty/ economic recession/ are they all correct?
We wouldn't normally say to pass over or tide over a crisis. Something can "tide you over" which means to help you get through a crisis. "Here's $100 to tide you over until your next paycheck."
We might say that a crisis will pass. "This food will tide you over until the crisis has passed."