Can you vent a plumbing vent through the wall?

Yes as long as the angle at which the pipe exits the wall and goes to the outside does not exceed 45 degrees.

Check the code for your area. Most I have seen go through the ceiling.

YES you can! you can even use a dryer vent to the outside.

One that vents one fixture

Provide air flow to or from drainage system and to protect trap seals from siphonage and backpressure.