Yes. The former is actually a flare-up of old chicken pox virus that's still wandering around the body.
Nope. Chicken pox is Herpes Zoster, and when it flares up, it's called Shingles and it sucks donkey cock. Herpes Simplex 1 (HSV1) typically causes oral lesions (cold sores), and is more common than HSV2, the virus responsible for most genital lesions. However, HSV1 can cause genital outbreaks, and HSV2 can cause oral ones, so it's a blurry line.
One of 'em, probably HSV1, can also cause blistering rashes on the body that are spread by contact. IIRC, high school wrestling is a breeding ground for that.
(Per my prenatal bloodwork, I don't have HSV1 or HSV2, so I have no idea why I have the vast herpes knowledge that I do, though it might have something to do with being really bored and reading medical texts as a teen.)