Yes. My Lab loves it. I can almost hear, "What's up Doc?" Though not all dogs like carrot. Try apple, if so.
Yes, carrot is fine. Although it's often not broken down by the dog's gut very well and can come out the same size it went in. So keep the pieces small.
I have been giving my pup a LARGE raw carrot everyday since about 11 weeks old. It's cold and hard and she loves it like a chew. It is her afternoon treat and I have my afternoon coffee at the same time As the poster said above, it does not break down. I see her "carrot poop" ever day at the same time.
I try to keep the pieces of carrot small and the total amount sensible for the dog's size. Slimy, orange poos are no fun and are quite alarming the first time!
What you are all observing is perfectly normal. Dogs can't break down the cellulose walls of vegetables. If you put the carrot through a blender or cook it then the dog can process the vegetable. So if your dog has a weight problem, then carrots will be an excellent treat for training.