The cinema and movie theater can both refer to the building as a whole, or they can also refer to the smaller rooms inside. In order to be as clear as possible when referring to a single cinema or theater, you can also refer to it as a "screening room."

First of all, where is your theater? I think the terminology is different in the UK and the US. Secondly, when is your project set? Is it supposed to be modern times? I think the commonly used terminology has changed a lot in the last forty years, as theaters have evolved from single-screen theaters to 24-screen "multiplexes." To answer for the US now, the "cinema" may have many "screens" or "theaters." A lot of them use the British spelling "theatre." The "showtime" is when the movie will start. Actually, the previews, or trailers, and advertisements start at the showtime, which can be annoying. A "performance" or "play" is usually a live show. A "function" is a meeting or a reunion or reception, just about anything that takes place at the "venue", except a movie, or "film". Look at this website for examples. http://cinergycinemas.com/corsicana/ I am going to see "Noah" tonight, so I will pay attention and save my ticket stub, and try to give you a better answer later. It is amazing how complicated an answer can be, when you think it is a simple question...