For a constant potential on the capacitor, there is no B-field and that is the case usually considered for this calculation. When charging a capacitor, the currents will generate a B-field and there is stored energy in that field (same as for an inductor). But once the charging stops, the B-field will "collapse" and cause currents to flow in the wires, dissipating that energy. Real capacitors will have some inductance and so will the wires feeding the capacitor and yes, you might need to include the effects if they are large enough (and they often do get included when analyzing circuits with real, not ideal, components).
Does that mean energy is stored in an electric field produced by a
point charge?
The classical self energy of a point charge is formally infinite and, thus, somewhat of an embarrassment.
Could you explain how energy can be stored in a magnetic field?
It's clear that energy is stored in a magnetic field so I'm not sure what you're looking for here.
When work is done, energy is converted from one form to another. Work is being done by the battery when the current through an inductor is increasing.
This is simply due to the fact the product of voltage and current is power, the rate at which work is done.
And, when the magnetic field threading the inductor coils is changing, there is a voltage across the inductor.
Thus, when the current through the inductor is increasing, there is a voltage across, proportional to the rate of change of current, and thus, an associated power
$$p_L = i_L \cdot v_L = i_L \cdot L \frac{di_L}{dt} $$
Further, when the current is decreasing, work is being done by the inductor on the circuit. So, the work done increasing the current (and associated magnetic field) is stored as energy in the magnetic field.
Best Answer
Energy stored in fields = the total energy required to assemble the fields
It takes energy to bring the charges to specific positions to assemble the field, and when you let everything go, the charges will just fly apart. The energy you stored in the field becomes the kinetic energy of the charges once you let them go.