Are you sure the leaving of 'presents' isn't indicative of any health problems?
I.e. he cannot go to the toilet in time or warn you he needs to etc. It may be worth getting him a checkup?
ETA: Can probably scrub the above. Adding this link:
http://www.puppymillrescue.com/Howtoadoptme.htm, just contains some limited general points (warning some distressing images/stories on site may make readers uncomfortable - though they aren't on that particular page). Beyond that GP's suggestions sound good.