And the owner of the coal should be liable - וליחייב בעל הגחלת
The question of וליחייב בעל הגחלת means that the בעל הכלב should be exempt from paying the half that the בעל הגחלת would have paid to a third party. There is no question that the בעל הכלב should be exempt for eating the חררה, for regarding the protection of his property the בעל החררה was not negligent.
Doors are generally susceptible to burrowing - סתם דלתות חתורות הן
תחלתו בפשיעה וסופו באונס cannot make one more liable than he would have been had it remained with the initial פשיעה.
Additionally תחלתו בפשיעה וסופו באונס is חייב only if the initial פשיעה was against the ultimate item which was damaged (but not if the פשיעה was for other items).
You can resolve that the mouth of a cow is like the courtyard of the victim - תפשוט דפי פרה כחצר הניזק דמי
The query regarding פ"פ was in a case where a non-responsible person fed the פרה. The גמרא initially assumed that only the taking needs to be from the חצר הניזק. The query was resolved since the גמרא establishes that the eating must also be in the חצר הניזק; proving thereby that פ"פ כחצר הניזק דמי.