My guess is that yes, you can; however, my mother always set them aside, not to be eaten. Any ideas?
They are not safe. An apple used in this manner is an aromatic, not to be eaten. I often use a combination of apple, cinnamon, rosemary and onion inside my bird. This stuff does not reach the necessary temperature to kill the little beasties that will hurt you. If it does, then you'll have one dry bird.
Depends. It increases the thermal mass of the thing to be cooked, making the inside (the apple) come to temp later than the meat parts of the bird.
If the apple gets hot enough, then chances are you've overcooked the bird. So pulling the bird when perfect results in an apple that might have some sort of contamination, although then the inside of the bird probably would as well (since they are touching).
For this reason, many chefs recommend cooking dressing (aka stuffing) outside the bird, either in the bottom of the pan or in a separate container. This allows you to cook the bird more quickly, and not to overcook it.
As an alternative to discarding (which seems a waste) there are any number of ways you could further cook the apple that would be better than just tossing it. Sounds like a nice addition to stuffing, or desert.
But you will need to cook it after you remove it from the bird. Stuffing is bad enough, and, in the case of stuffing, you can raise it to a high temperature before you put it in the bird.
Only if the apple gets cooked to a bacteria-killing temperature. In chicken's case, I believe the USDA/FDA recommends 165, which is insane. 150 should be fine.
Of course by that point you may as well make your roast chicken into shoes.
Discard the apple.
There's a little bit of information at this post: What do I need to know about temperature and food safety?
In brief, to be certain, test with thermometer that it's over 140F and has been for 10 minutes (hence the 150F or so suggestions here).
That said, the inside of your bird probably shouldn't be getting that hot; cook the bird to desired doneness, the apple is a sideshow item.