"Hate speech" isn't just hateful speech, it's a specific term with a specific meaning. Being a nazi isn't an inherent characteristic of a person, it's an affiliation or ideology that they consciously choose.
No when it's a label deliberately misapplied to run of the mill conservatives. That's defamation with the purpose of generating hate against those people.
Sure.
But the overwhelming majority of people called "nazis" by their political opponents have objectively not chosen anything remotely of the sort.