Yes. There are many diffrent strains of the Herpes Simplex virus and some are carried by dogs. Usually they can't be passed from human to dog or vice versa though.
Yes they can. They can catch herpes simplex, but it usually isn't so bad for them. Though if they breed it will end up either killing the puppies or making them go blind. Though not many get sores on their mouth.
Not specifically. Dogs do not have lips. Dogs can get Hepatitis and have sores around their mouth and anus. Biting their own mouth and scratches are more common. Be sure to check with a vet.
Yes. They can get cold sores. And the best thing to do about it is keep it drinking water and go to your local vet. They have things for that. But be careful! Cold sores are contagious! It takes about 10 days for it to heal. The only time that you need to go get medicine