RAW: they are persistent bonuses in play from monsters
I'm not sure the designer's intent, but I'd say per RAW the answer:
Does it count as a "card in play" to discard to a spider's overrun?
No. You can't not "discard" monster cards, they can never go into your deck.
Does it count as a "non-item card in play" for the World Spirit?
While I feel the intent maybe no, RAW I'd have to say "yes".
They are definitely not items (no Item keyword or grey border).