I read this factoid (actually, it was a whole article about the Thanksgiving turkey industry) in a newspaper a couple of years ago around Thanksgiving time:
Thru selective breeding, turkeys that are bred for Thanksgiving are so fat that they can not mate (without some kind of aid, presumably).
Another factoid from this same article was that “There are only three (I think) companies in the Thanksgiving turkey breeding industry.”
Any verification on these?
Was this the article?
No info yet on whether it’s true, but AFAIK it is. But it’s not because of fat–it’s because modern turkeys have been selectively bred to satisfy the demand for an extraordinarily high percentage of white meat in the dressed turkey carcass, and this makes it difficult for the male turkey to get on top of the female and copulate properly with her. I guess you could say that his breasts get in the way. 
Here’s the USDA on the subject.
Looks like the answer to your questions are “yes” and “probably”.