Their website has details about arranging a film shoot. They definitely partner with nonprofits on occasion. If you were just looking to take footage of the inside of the theater, LA Conservancy does yearly screenings that allow people inside to see movies there and wander around, though you just missed going by a day.
Maroon 5 did their "Moves Like Jagger" video here so I highly doubt it. But there may be circumstances they will allow it. I would check their website or contact them for details.
As a setting for a little movie, not for free. It's a popular filming venue. During a Last Remaining Seats screening in the summer, it's expected for people to film this beautiful theater.
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