To "get someone's back up" means to put them in an adverserial position, or have them be on the defensive. In this case, the "head" refers to the person in charge of the school. The friend is being advised to discuss the matter of her child's treatment at school in a calm, non-confrontational manner.
"Ned, Ned, wets the bed" is a rather rude taunt, and only people with a mean streak or childish sense of humor would find it funny. That's the "rather pathetic level" being referred to.
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