Must chess pieces be within a square? In lightning chess. must pieces be moved so they are completely within a square? If not then how much overlap is allowed? If my opponent breaks this rule do I stop the clock and call the DOP?
Yes, pieces need to be properly placed. You can either tap the clock back and ask him/her to adjust his/her piece (which is what happens in many local level tournaments). It is generally better to check with the arbiter before the start of the game on what is the rule for handling such a scenario in their tournament since calling the arbiter mid-game can get you penalised. 

The official FIDE site says that pieces can be adjusted only in each player's time. 

I would suggest use your judgement and good sportsmanship sense and not call for the adjust for a small overlap but that is my opinion.