On that question, I have no strong insights. Based on my scientific background though, I would say that the shape and behavior of the earth’s magnetic field might be influenced by the strength (or lack thereof) of the sun’s magnetic field, and by the solar wind. But the overall strength of the earth’s magnetic field would typically only be a function of the metallic constituents of the earth’s interior and how and where they are flowing/shifting position.
Agreed. Depends on how strong the sun’s contribution is near the Earth. The charged particles flying off of the sun may have a greater impact than the sun itself.
Extending the idea, if there is a relationship between solar flares and the strength of the sun’s magnetic field, and there is a minimum of flares at any point in time, that would mean that the sun’s magnetic field projecting itself through space would be reduced as well.