Wooden storm window repair and painting
Since you've reglazed, there aren't many options for re-clamping the corners, so... do repairs with a two part wood filler ("bondo" in North America).

Regarding the primer, try to find out if it was latex or oil based putty (tip: if it dried fast, it was latex; if it can still be imprinted with your thumb after 3 weeks, it's oil), and match your primer to that. Your painter is correct -- you should overlap a 16th onto the glass.