As Julia said, there is nothing saying a doom token is not added when you use Naacal Key.
As for the phase issue: the rules say when a gate opens ON you, you're immediately drawn in and delayed. If you assume this occurs with the Naacal Key, then you won't enter on the Arkham encounters phase: you'll enter immediately when you use the Key and will be delayed. But, as I've said, this renders the key so horrible, that I don't play that you're delayed on the grounds of theme, in the same way that I argue that a Nightgaunt doesn't cause you to be delayed when it throws you through a gate.
Unclejo: I don't think Gate Box is extremely great. It's about worth the $4 list price.