Agreed. And how much does the sun and charged particles induce the Earth’s magnetic field?
You can compare the earth's field lines to the chart of the field lines of a solenoid and they are basically identical:
Now, with a solenoid, just the current moving through the coils creates that kind of magnetic field on its own. The earth, however, is rotating. If you have a rotating conductor exposed to an external magnetic field, then that field would induce a current in the conductor, and if the conductor were arranged correctly, the current passing through it could form its own magnetic field just like a solenoid. So it's possible that the whole magnetic field of the earth is a consequence of the sun's magnetosphere combined with the composition and motion of the earth.
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.