How does a self-centering rudder affect left-turning tendency?
First, "adverse yaw" refers to the increased drag of a down aileron (and reduced drag on an up aileron) -- which mean that as you deflect the ailerons to roll left (in order to turn left), the adverse yaw makes the nose yaw right -- exactly the opposite of what you want. Therefore you balance any application of aileron with just the right amount of rudder to counteract the adverse yaw. This is called "coordination".

For any control surface (like rudder or aileron) there is a small "dead zone" around neutral, where movement of the surface does not have an appreciable aerodynamic effect. Those narrow strips reduce the size of that dead zone and improve control. Therefore they help, not hurt.

What you're talking about is what the FAA (and others) call "left-turning tendency". When flying straight and level, the engine power affects the amount of rudder required, and significantly so. At high power more right rudder is required, and especially so at low speeds like climbing. At low power settings less right rudder is required, especially at high speeds -- so in a descent many aircraft require left ruder to straight. (Of course this is most applicable to single-engine airplanes where the fin is in the propwash.)

To help the pilot with the forces required, surfaces have trim tabs -- see Why Pitch Trim Up/Down &amp; Roll Left/Right switches on yoke or control stick?

(Incidentally, many older small English engines like the dh Gipsy engine, https://en.wikipedia.org/wiki/De_Havilland_Gipsy_Major, turn the opposite direction so airplanes like the Tiger Moth that use that engine have a right-turning tendency and require left rudder.)