They are a conceptual convenience, but only insofar as they allow easy tracking of the integrals needed for the complete calculation. The diagrams and the integrals do represent reality however. The evidence is in the accuracy and precision obtained from the calculations in their description of reality. For instance the description of the lamb shift in the hydrogen atom.
The radiation from a black hole also indicates the validity of the reality behind the integrals represented by the Feynman diagrams. The radiation amounts to pair production out of the vacuum, where a net capture of antimatter occurs.
http://www.freerepublic.com/focus/fr/605554/posts?page=70#70
Of course.
The calculations are part of a theoretical description. Integrals are mathematical entities, with no necessary link to any physical phenomena. The important aspect of this is the theory used in desribing the phenomena.
You could use a purely numerical approach in calculating these effects without ever utilizing a Feynman diagram or an integral.