I block my opponent's 3/3 beast with a face down Chromeshell Crab. After declaring blockers, I morph Chromeshell Crab, and switch with another creature my opponent controls. Does the 3/3 beast still take damage, and does Chromeshell Crab along with the beast still die on my opponent's side?
No and no. A creature changing control also removes it from combat. The 3/3 is still blocked, but the Crab is no longer blocking. The 3/3 therefore will deal no combat damage due to not having any blockers anymore (unless it has trample or some other ability to still allow it to do so), and the Crab will neither deal nor receive combat damage due to not being in combat.
506.4. A permanent is removed from combat (...) if its controller changes, (...). A creature that’s removed from combat stops being an attacking, blocking, blocked, and/or unblocked creature. (...)
510.1c A blocked creature assigns its combat damage to the creatures blocking it. If no creatures are currently blocking it (if, for example, they were destroyed or removed from combat), it assigns no combat damage. (...)
Former Rules Advisor
"Everything's better with pirates." - Lodge
(The Gamers: Dorkness Rising)
"Any sufficiently analyzed magic is indistinguishable from science."
(Girl Genius - Fairy Tale Theater Break - Cinderella, end of volume 8)
510.1c A blocked creature assigns its combat damage to the creatures blocking it. If no creatures are currently blocking it (if, for example, they were destroyed or removed from combat), it assigns no combat damage. (...)
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules