The non-member spouse.
As far as what to do if the member dies, in general I think you just leave them as they are and the system will take care of it. Depending on the kind of record it created for the spouse the record may just sit there as a spouse record forever, which is fine since they are still a spouse. Some kinds will eventually be recorded as deceased non-member, if there was a birthdate with the non-member record.
I think the most important thing to do before recording the death information on the member is to do a quick review to make sure everything is as accurate as possible since you'll likely be the last person to touch that record in this life.