Direction of Earth’s poles relative to the Sun’s movement around the Galaxy
There is no difference in the number of cosmic particles.

You seem to imagine that there are static clouds of gas in the universe that the sun is moving through.  This is not the case. The gas in the galaxy is orbiting at roughly the same speed as the sun.

The sun does have some proper motion relative to the interstellar medium, but the interstellar medium doesn't penetrate the bubble formed by the solar wind.

There is no dragging. The gravity of the galaxy affects both sun and planets, we all move around the galaxy together. There is no shifting of the plane at all.

Interstellar particles are found in ice cores, a paper suggests that variation in the rate of cosmic radiation may cause hydromagnetic shocks that could transport metal rich dust into the inner solar system. No 30 million year periodicity is mentioned.