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.
FIDE rules are silent on this subject. In standard chess players would typically say "j'adoube" and adjust the piece accordingly if there was a slight overlap and a player putting a piece half on one square and half on another would either immediately readjust or his opponent would challenge him to put it on the right square.
All of this becomes a bit more fraught at blitz. If the piece is touching a line or marginally over the line then I would suggest you either adjust its position on your time if it disturbs you (you may not touch the pieces during your opponent's turn) or ignore it.
If it is well over the line, close to half and half then you could on your turn adjust the piece to the square which is better for you but obviously you need to be prepared for reaction from your opponent. A better approach would be to call the arbiter and claim the game on the grounds of an illegal move. e3 and e4 are both legal moves. e3.5 is not. Note, for comparison, that e8="upside down rook" is an illegal move.
Edit: I had this happen to me in a rapid competition a week ago. My opponent played Rb6 - g7+ which was a devastating game changer. Rb6 - g6+ would have been a boring move which wouldn't have changed much. I immediately stopped the clocks, called the arbiter and claimed the game.