QWhen I play this follower, will it trigger abilities that mention Puppets, such as Noah, Vengeful Puppeteer's ability?

AYes, it will.

QCan an ability that references Neutral followers on the field also reference this follower?

ANo. Having the additional card name Puppet doesn't mean this follower also has the same card information as the Puppet token, such as its cost and class.

QWhat does "While this card is on the field, its name is also Puppet" mean?

AWhile this follower is on the field, it has the additional card name Puppet, and any ability that references cards named Puppet can also reference this follower.