Dog's can't chew gum. Most of the time they'll swallow it as soon as you give it to them. I don't think it will make them sick in small doses but I can't imagine it would be very good for them either. I'd try to avoid giving gum to your dog as much as possible.
No they cant, at least not for long. If they do swallow this which is actually quite unlikely they would most likely digest it much like a human causing no harm, unless they swallow multiple pieces.
