Does Divine Judgment recycle trash and banish cards?

No — Divine Judgment only touches the card types it names: units, gear, and runes on the board, plus cards in hand. Cards in the trash or in banishment are not among the "rest" and stay exactly where they are.

The same logic is why hidden cards are unaffected: they aren't units, gear, runes, or hand cards, so they're neither chosen nor recycled (FAQ 7410).