Should I caulk window gap behind casings?
No. It would take a massive amount of caulk and you'd have a real mess. Or yes, if you consider spray foam "caulk", which it is in a sense. Water sealing is done outside. The seal(s) at the outer face of the wall handle water and and air intrusion concerns, ideally. Modern techniques for that involve flexible tapes and housewrap membranes.

Traditionally that void would have been filled with fiberglass insulation. In recent years we've taken to using spray foam from disposable cans, which provides both insulation value and a nice air seal, and it offers the added benefit of stability for the window. It should not be relied upon for water management, however. 

Also, be sure to use the "minimal expansion" variety, as the original version is very likely to bow your window frames (jamming the sashes) and will end up all over everything once it squeezes out. Once urethane (foam, glue, varnish) dries on something it will be there a very long time.