The difference between zij and ze, is the same kind of difference between jij and je ("you" and "ya"). IOW, Ze and Je are very colloquial and a natural change in pronunciation in normal conversational speed. Using zij and jij would be like using "you" vs. 'ya' in very careful speech.
When he said is a listening exercise, it doesn't mean that you don't have to write. What he is saying is that you have to write what you hear (in simple words) that's why is a listening exercise: Listen carefully to write down well. In this case grammatically both option are correct. But obviously the guy of the record pronounce just one of them.
Lol Makes me laugh at how alike German is. :D Sie haben das/es getan vs Zij hebben het gedaan. I love it. No wonder I understand so much Dutch without ... actually understanding it. xD Ze zijn daar seem to sound the same as sie sind da, to me; am I doing something wrong?