Can a child from a non-Buddhist family attend a class in Buddhism?

Yes, the child can, provided the parents give their consent in writing.

Can a parent request the school to teach Buddhism under religious instruction, to a child of theirs?

Yes, they can in the case of a government school.

In schools to which BESS provides its services, are teachers sponsored by temples?

Do your teachers hold blue cards?

Yes, regular teachers and teacher aides do have blue cards.