Can Air Ionized by an Electric Field Transmit Electricity from Another Source at the Same Time?

Playing with lightning is dangerous!! Either people and/or equipment will often be sacrificed when one discharges kilovolts of electricity. One must be very careful with a design to choose sacrificial material in sheltered places, instead of people, when working at these voltages.

The switch you are contemplating is called an trigatron. It is one of the types of thyratron devices that allow for the control of massive amounts of current used in pulsed power.

(From the Wikipedia link)

{Notice the wire mesh surrounding the vessel. It is there to contain an explosion when metal fatigue happens.}

The spatial geometry of your multiple electrodes will be very critical factors in determining how much energy gets applied from your high to low voltage circuitries and vice versa.

This is not a lossless method of power transfer. Generating and certainly maintaining a plasma requires power consumption. Usually the switched power is orders of magnitude greater than the losses to make the plama. When these devices are used in pulsed power applications, one has the added advantage of brief time intervals of operation.

In addition, background ionizing radiation can and will at random times prematurely trigger a plasma path.

Lastly, in theory your idea has some validity but the safety concerns to you, others nearby and the rest of your circuitry should prevent you from trying this.

The quick answer is that this won't work.

The electrical conductivity of a plasma is a function of available charge carriers, i.e., the degree of ionization in the plasma. Although an electrical corona at STP does pass a small current, it's typically of the order of microamperes to milliamperes with applied potentials of thousands of volts and surface E-fields above 27 kV/cm. The effective "resistance" of a weakly-ionized cold corona plasma between the plates of the order of 10e9 ohms/sq-cm - far from being lossless. Thermal plasmas (high-current arcs) are MUCH better electrical conductors. But even a heavily-ionized thermal plasma is not nearly as efficient an electrical conductor as common metals.

Use a wire...