Yes a witness box would be good example of a "stand". Its the place in which a person stands in a law court when they are being asked questions. But with people being lazy these days they tend to have seats, so normally just the first little bit remains "standing" at the "stand". An yes call someone to the stand just means they are "requesting" a person to come up to that stand.

Yes. The "stand" is the witness box.

So in this video, and in general, calling someone to the stand is when, during a case in court, a person is requested to sit on the stand and answer questions regarding the case at hand. In this video, the accused (school) is asked to sit on the stand to answer questions to defend himself/itself. Does that make sense to you?