Need egy here or no?
It would sound very strange without.
For me, it means the same with ot without 'egy'
And it didn't accept "fura" instead of "furcsa"
You can report it. They mean the same.
Why is 'mozi' marked incorrect?
Mozi is the movie theater, not the movie itself.
That is generally true, but occasionally it is used for films, as well. Maybe in a more general sense, as in "a good movie experience".