Depends. White scar-like lesions around the genitals & anus could represent any number of skin conditions... Molluscum contagiosum, condyloma latum (syphilis), and herpes are examples of some contagious conditions while other conditions (like lichen sclerosus) can be progressively disfiguring. It is important to have you partner checked out by a physician like a dermatologist to rule out concerning causes.
It's possible. It could be herpes scars or it could be something else. There are blood tests which can be done to show if someone has been exposed to genital herpes. Both of you should have the tests done because many people have herpes who don't know about it.