In the span of a single lifetime, none.
Over the course of human history, one theory holds that the slowly-changing variation in the direction of Earth's axis may be responsible for some long-term climate changes. This is a tough one to prove or disprove, as other factors (including the varying eccentricity [ovalness] of Earth's orbit around the sun) may also be responsible.
Earth's precession
precession
precession
Procession of the Equinox
I think you are referring to a phenomenon called Nutation.
The Earth is a spinning top (gyro) with a different moment of inertia about the spin axis than the other two axis. Therefore, it acts like a spinning top with both precession and nutation. Precession of the polar axis (relative to a 'fixed' distant star) forms a cone. The precession period (~26,000years) is the time it takes for the cone to be traced. Nutation occurs normal to the precession cone and has a much faster period (~18.6yrs).
Impact craters.
Lots.
The earths rotation effects whether it is night or day (so you'll see the moon at night).
Due to the earths precession (How the earth 'wobbles' on its axis) the pole star changes over time. Though this takes many, many years
Which Earth cycle are you referring to (there are many). If you want us to answer your questions you must be sure your question is as precise as possible.
In the last couple of million years, the Earth has experienced dozens of ice ages. The last one ended about 15,000 years ago. We don't know if they are related at all to the Earth's "precession", which is a 26,000-year-long "wobble" in the Earth's spin. There is no reason for precession to have any effect on Earth's climate, since the 'tilt' of the axis remains constant throughout the cycle of precession.