RE: gettinng a downed bird
Over here the advice is fetch the bird but leave the gun behind. Few people will have a problem with you retrieving shot game from their property.
Those that would have a problem cant do much about it as trespass is not something you can be prosecuted for over here.
Armed trespass on the other hand is a serious offence and could involve jail time and certainly the loss of your gun licesnse.