For every action, there is an equal, opposite reaction. Therefore, does the space station move in the opposite direction each time an astronaut pushes off from a bulkhead?

Yes, ever so slightly. But the astronaut must then soon contact the opposite side of the station, imparting a force in the other direction.

Yes, it would. Since the mass of the station is so much larger than the mass of the person, it accelerates much more slowly, so that it isn’t really noticeable.

Like @thorninmud said, once the astronaut contacts another wall, the station is accelerated in the opposite direction, returning it to the original velocity.

As indicated above, every action does indeed have an equal opposite reaction. However, the ISS, along with other space vehicles, has an on-board Attitude and Articulation Control System (AACS) which is designed to compensate for any type of inertial anomalies that might occur during deployment.

Sure, just like Earth itself.

More applicable to say that the Space Station is in orbit (zero G). Action-reaction is harder to observe on Earth (one G), even in a vacuum chamber.

Corrections for orbit are checked constantly and with absolute precision. The occasional pressure from a space walk is nothing compared to the constant stress of Earths gravitational pull. http://en.rian.ru/science/20110701/164940749.html

Along the same lines of this question, what if everyone in the world all jumped at once? There’s a fun answer to that one in the linked article.

The space station will move a tiny bit, but the distance it moves is so imperceptible due to its relatively greater mass than the astronaut. Think conservation of momentum.