Assuming all options are viable (i.e. that Billy fits in the supply closet) I’ll go with this one.
He wants to play hide and seek, not cause undue mayhem and possibly get into too much trouble. Plus, all the classrooms I’ve see in the US have the door close to the teacher’s desk. I’ll give Mrs Kringle the benefit of assuming she’d realize the main door is being opened: chalk on blackboard doesn’t make that much noise, and she must have at least some peripheral vision. Besides, she’s a grade school teacher so she must constantly be on her toes in case someone who should be in class wanders out and someone who shouldn’t be there wanders in.
So that rules out Option 3.
I’m tempted to go Option 2 simply because that glint in the children’s eyes might suggest a collective prank on the making. Which could start with Billy under the teacher’s desk.
But the text specified Billy wants to play hide and seek and no one in the classroom seems to be snickering, so they either didn’t notice him move or can’t see that he’s not where he’s supposed to be.
So I’ll go with Option 1: supply closet.