No, it is impossible to breed and pass down genetic abilities. The ability that is generated for the offspring is random, and nothing in the game can change the odds of the abilities.
Think of it like this, if you were able to pass abilities through Breeding:
Shedinja and Voltorb are both from the Mineral Egg Group, so they are cabable of breeding. You breed them together. Shedinja's the father, Voltorb's the mother. Your possible result is an offspring Voltorb with the ability Wonder Guard.
Shedinja has five weaknesses (fire, rock, ghost, Dark, and Flying.), whereas Voltorb only has one, which is Ground. The passing of abilities would throw game balance out of control (and make the Wi-Fi community spammed with Wonder Electrode, of course. =P).
So in conclusion, no, this is not possible. While it would be interesting, it just wouldn't turn out well, seeing as the communty would heavily abuse this feature.
EDIT: I apologize for this post going off-topic. I felt I should englighten you as to why it doesn't happen.
EDIT 2: Yeah, so I just figured out that Shedinja cannot normally breed, due to its lack of a gender. You get the point though, right? xD