Wash basin in Garage Drainage Question
Yes, you can go through through a wall with ABS. In certain situations we put in a sleeve for the pipe to go through (ie concrete walls, block walls, foundation walls, etc). If its just a drywalled stud wall then no sleeve is really necessary.
Yes, you do need a vent. This allows air into the plumbing system. Ever hold your finger over the end of a straw and try to get coca-cola to drain out?
Yes, you do need a p-trap. This keeps sewer gases out of your dwelling. If you didnt have one you'd swear "it smells like open @$$ in here!"
So, like Mike said, "Yes to all three"