JAQEN H'GHAR, FOLLOWER OF THE RED GOD: A NAME GIVEN
Does Jaqen also count as having the name of the unit whose Influence he copied?
Yes. Replace all instances of that unit's name in the Ability with "Jaqen H'ghar"
Can Jaqen replace his Influence Ability with an Influence Ability from one NCU that had, previously that turn, lost its abilities?
No. There's no ability to Jaqen to copy.
Follow up: If the NCU that had its influence copied lose its abilities, does Jaqen lose his Influence as well?
No.
Does Jaqen also copy other effects above the “Influence” effect within the same named ability?
Yes. Everything under the name of the ability is the same ability that is the Influence ability, Official FAQ also points out to replace original unit name with Jaqen in all instances.