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2"
No and yes. Amendment (not from a professional plumber!): If you were simply running the sink drain into the shower drain and if there was not a reduction in pipe size, I would think you would be able to do that. I think the reason you need a trap for the sink has to do with code that requires a trap be no more than 24" from the basin to the trap arm. That being said, the pedestal sink in my house does not have a trap until after it drains below the floor.
Depends on the mateiral you already have (PVC) Galvanized nippe and brass P trap or rough brass connectiong to the trap and then of course what size of drain basin, KS, Slop Sink
Most of the time it is 4 inch.
downstream of a closet bend within 8 ft for three inch pipe and 10 feet for 4 in pipe at least half the size of the drain... unless it is stack vented then it would be full size of the stack right