1.) Is the right of privacy implicated when a woman has an operation performed on her in a public place (hospital or clinic) by a person she probably barely knows"

I think "The Right of Privacy" is only implicated if the woman would have the abortion done by a private physician. Since it is being done in a Public Hospital, the woman will not have "The Right of Privacy." It is her risk of doing the abortion by a person she hardly knows, so I think "The Right of Privacy does not apply in this situation.

2.) Which seem closer to the core of an intelligible privacy doctrine--an abortion or consensual sex between adults"

I think the core of an intelligible privacy doctrine would apply more to consensual sex between adults, because I think this is more private than wanting to have an abortion. If you were having an abortion...