If a Death Note user specifies, along with the actual target, the name of a person whose face the user doesn't know, will that person live?
The person would live in many cases. That rule indeed says that if multiple people are specified, that they all would end up having a heart attack, but it does not circumvent all the other existing rules. You would still need to have the person's face in mind while writing and the name should be the person's real name and not a nickname. The only thing that rule says is that all other details regarding the first person's death are just ignored.

For example if you write


  Person A stabbed to the heart. Person A would hug person B and would then kill person C. Person D would become jealous and would stab person A.


the Death Note would read it as


  Person A stabbed to the heart. Person A would hug Person B and would then kiss Person C. Person D would become jealous and would stab Person A.


or


  Person A
  Person B
  Person C
  Person D  


So unless you had the specific person's face in mind while writing the name or unless the name was fake, the person would die from a heart attack.