what is the day mode for? And is there any way to turn on a ringer for all phones if we need it (say all the secretaries are gone? On our old TIE system we set a "night ringer" which rang incoming calls over our paging system. Is there a similar thing with the K3 systems?

Yes and yes. I had to look the bell part up. Your paging system will need a contact activated warble. basically the system has a loud bell dry relay contact that can be activated on incomming calls and what not.