Co-Op Rooms are unlocked in single player by simply finding them on the map.
Sword of the Gods:
After you collect each of the 42 Mark of the Gods, the Sword of the Gods will unlock and any melee damage done is doubled.
Environmental Exterminator - Destroy a Spider using spikes (General).
That's Teamwork - Destroy an enemy using a Kamikazer (General).
Don't Go Alone - Complete any Co-op Challenge Room (General).
Enthusiast - Collect 10 Marks of the Gods (Single Player only).
Two Birds - Smack a bomb so that kills two or more enemies (Individual).
Survivor - Kill 20 enemies in a row with only 1 health left (Individual).
Overkill - Destroy six enemies with one Beam Shot (Individual).
Arcade Master - Complete any Arcade Chapter below the target time (General).
Untouchable - Defeat any boss without taking damage (General).
Collector - Collect all Marks of the Gods (Single Player only).
Millionaire - Collect 1 000 000 Doubloons (Single Player only).
Grand Master - Complete the full game (Individual).