Do I need to copy old entries to my new logbook?
My recommendation - don't copy over anything from your old logbook.  Just start your new logbook with your next flight or  other item that needs to be logged. You will simply have more than one logbook.

You don't need to carry your logbook with you while you are flying.  Pilot cert and medical is all that is required.  If you need to produce a record verifying that you are qualified for a particular type airplane, operation, etc., you can produce your record (logbook[s]) at a later time.