Can I (or a student or teacher) take an iPad to an Apple Store to ask for technical support?

Probably not. Apple does not provide support for devices that are managed by a Mobile Device Management (MDM) server.  The person at the Apple Store will quickly be able to determine that the iPad is owned by and managed by the Diocese of Brooklyn and so they will probably not want to help you.  However, if it is just a software-related question, they will probably do their best to answer it.