What is the angle between the equator of the sun, and the plane of the Earth's orbit
The plane that contains the orbit of the Earth is known as the "ecliptic". The rotation of the sun is tilted by 7.25 degrees to the ecliptic, and this value does not vary over time.

The rotation of the Earth is also tilted, by 23.45 degrees to the ecliptic, it is this angle that causes seasons.

An old paper Position of the Sun's axis describes how the Earth crosses the solar equator on June 4th and Dec. 6th each year.

Nasa has a factsheet, that has this and other information about our nearest star.