Why does the impeller and turbine have vanes in opposite directions?
There are several parts to a torque converter (TC):
- Transmission Fluid
The housing it attached to directly to the engine. It spins at the same speed as the engine (obviously). Attached to the housing are the fins of the pump. The pump is a centrifugal type pump. As it spins faster, more fluid is moved through the torque converter. The turbine portion of the TC has blades, which you pointed out, point in the opposite direction of the pump blades. They are set opposite so that they can catch the oncoming fluid which is being forced at it from the pump blades. This causes a directional change of the fluid, which in turns causes the turbine to spin. The turbine is the portion of the TC which is attached to the transmission itself.
Why does the turbine have more of them?
Every manufacturer is going to build their TC's different, this includes the vane count on the pump and turbine. This means the turbine will not always have more than the pump will. The manufacturer's design will also include what the pitch is on any of the vanes, how closely they are run together, overall diameter of the TC, and many other factors. While the overall concept and work ability of these are the same, the differences are distinct.
Why does the stator have to change fluid direction anyways?
The stator's main job is to increase the efficiency of the TC. The TC overall is a fluid coupling device. Without the fluid, the pump would turn, but nothing would happen. With forcing the fluid through the TC, it becomes coupled, and the force is transmitted to the transmission. The stator is there to make the TC not just a fluid coupling device, but also allows it to become a torque multiplier. This allows low torque areas of the torque curve (low rpm areas) to generate more torque and allow it to be transferred through the transmission and move the vehicle. Without the fluid change, none of this would be possible. Physics dictates, in order to move an object you must apply force to the object. The directional change of the fluid provides the force.