Yes they can. The theory of gravel eroding their barbels is nonsense. I have kept many corys in gravel with no issues. It is believed to be caused by ammonia now by most people.
Depends on the gravel...some might be too sharp, but typical aquarium gravel is smooth enough IME. I have some on Eco Complete which is larger grained too and they're fine. I'd worry more about poor water quality causing barbel issues than the substrate.
I had kept my Cories with gravel for 2 years and they never spawned. They also weren't that active and to be honest their barbels were smaller. As soon as I changed to sand they went crazy. They have been in sand for 2 years now and they are happier. They now have produced tons of Cory fry and are still spawning once every 2 or so weeks. I have sand in all my Cory Catfish tanks. While none of my Cory Catfish died in gravel I think they tend to be happier with sand. It depends on the type of gravel you have them in.