Ha'tasrit kotev et atzmo.
I never heard this word before. Does it refer only to the script of a movie which the word תסריט suggests or can it be a script for a play as well?
Yes. Just the script of a movie.
For a play it would be מחזה.
By the way, how in English do you distinguish between הצגה and מחזה?
לא, לכתוב תסריט זה מאמץ גדול!
do you need "et" if reflexive?