Why *remedy* means school term break

"Break off" has two meanings that I can think of off the top of my head. The one that you refer to in your question can simply be replaced by "cancel" in all instances.

He broke off our plans <-> He canceled our plans

The second "break off" is a more literal phrase and it implies that something was removed from a greater whole.

"He broke off from the rest of the team." means he left the rest of the team and went off on his own. Likewise, "He accidentally broke off a piece of the vase." means part of the vase was removed from the vase.

Not really, 'in school' is perhaps more common American English while 'at school' is more British but both are equally 'correct'. Similarly an American would probably say 'in college' while a Brit would say 'at university'.

In tends to be used for institutions, so you are 'in hospital' rather than 'at hospital' but 'at home' not 'in home' - although you might be put 'in a home'

It's just one of those things!

edit: there is perhaps a slight subtle difference that 'in school' means they attend school - as opposed to having finished school, while 'at school' means they are there now.
So "are your children in school" = are they under 16 or 18 ? But "are your children at school" = are they at school today or are they at home.
(but that's from a BE perspective)