(N/A) Yes,the earth's magnetic field changes with time. It changes appreciably over a time scale of a few hundred years.
$(b)$ The earth's core contains molten iron. At the high temperatures present in the core,iron is not ferromagnetic,and therefore,it cannot be the source of the earth's magnetism.
$(c)$ The 'battery' or source of energy that sustains these currents is believed to be the radioactivity in the earth's interior.
$(d)$ Geologists can infer the history of the earth's magnetic field by analyzing the magnetization of rocks formed during their solidification in the distant past.
$(e)$ At large distances (greater than about $30,000\; km$),the earth's magnetic field is distorted by the solar wind and the interaction with the ionosphere,where the motion of charged particles creates additional magnetic fields.
$(f)$ Yes,even a weak magnetic field of $10^{-12}\; T$ can have significant consequences over the vast distances of interstellar space,as it can cause the deflection of charged particles moving through it.