They will do a bag check at the entrance. If it's small enough you can keep it under you seat otherwise you will have to keep it with them
It will be inspected at the entrance, and you can always put it under the seat.
You can bring a backpack in after security inspects it but no, there is no place to check it. You'll have to keep it in your lap and given the tiny seats I would not recommend it.
I don't know but I guess it's rather difficult because of the small, cosy size of the theatre.
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