Most schools with forced air ventilation should already have such a capability, if they bothered to look. If not, if would certainly be cheaper to have one installed than to leave the windows open. But of course, it may be easier to do one than the other bureaucratically.
This is patently untrue.
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That way outgoing air would warm (or cool) the incoming air. If a Hepa filter could be added to the incoming air fan it would make an ideal unit to put in the window at fairly low cost. Since they are CPU coolers they could probably be made speed adjustable also.
Countercurrent exchange is the way to go with a scheme like this, so you'd want a series of pipes at varying temperature.