Except for the listed exception, the token created by Abyssal Harvester's ability copies exactly what was printed on the original card and nothing else. It doesn't copy information about the object that card was before it was put into that graveyard.

The token is a Nightmare in addition to its other creature types. This is a copiable value of the token that other effects may copy.

If a card copied by the token had any "when this creature enters" abilities, the appropriate token also has those abilities and will trigger them when it's created. Similarly, any "as this creature enters" or "this creature enters with" abilities that the token has copied will also work.

Abyssal Harvester's ability will exile all other Nightmare tokens you control, not just Nightmare tokens created with Abyssal Harvester's ability.