Translation:I am angry because the cinema is not open.
It's because níl is the negative of fuil (dependent form of bí) so it comes first in the phrase. Oscailte is the adjectival form of osclaíonn; Because it's an adjective, it comes after an phictiúlann. So that's how we get níl an phictiúlann oscailte :D
Your native speaker friend wasn't technically correct about the difference between táim and tá me, as "I'm" is considered less formal than "I am", and there is no such distinction between táim and tá me.
With regards to this exercise, as this is an Irish to English exercise, the only time you get to give an answer in Irish is when it is used for a "Type what you hear" exercise, which is meant to measure your ability to hear the actual words spoken. As she doesn't say Taim feargach mar nil phicturlann oscailte, you have not accurately heard the words spoken, so your answer was rejected.
If you were asked to translate "I am angry because the cinema is not open", then Taim feargach mar nil an phicturlann oscailte, with the definite article, would probably be accepted (though Ta fearg orm mar nil an phicturlann oscailte would be better).