The fact that theremins use heterodyne mixers has nothing to do with RF. The 'antennae' are not antennae in the classical, RF sense. The capacitance explanation is correct.
Capacitors and Theremin 'Antennae'
The simplest type of capacitor is a parallel-plate capacitor. That means the capacitor consists of two metal plates separated by some material called the dielectric. The equation for the capacitance of such a capacitor is C=εA/d, where ε is the permittivity of the dielectric (ε≈8.8541878176..×10^−12 F/m for air).
When you are operating a theremin, your hand is one plate (your hand is effectively grounded), the antenna is the other, and the air between the two is the dielectric. As you move your hand, you vary the capacitance between ground and the antenna. Both hands will affect both antennae, as they act like two plates in parallel, increasing the total area.
The two antennae are at right angles because that reduces the impact your left hand will have on the right antenna and vice versa. For example, as you move your hand up and down above the volume antenna, it maintains a relatively constant distance from the pitch antenna, thus it's contribution to the overall capacitance is constant (and small).
Theory of Operation
Note/Update: Please refer to FredM's Answer for a more detailed description of the oscillator.
Both antennae capacitors are part of two different, complex active LC oscillators. The 'L' refers to inductors, which store energy in a magnetic field; the 'C' refers to capacitors, which store energy in an electric field. In an LC oscillator, energy is constantly flowing back and forth between the two, changing from electric potential to magnetic potential.
The frequency of the pitch oscillator is beyond audio frequencies, so it can't be directly used. The theremin has a third oscillator that operates at a fixed frequency. The pitch oscillator and the fixed oscillator's outputs are fed into a heterodyne mixer, resulting in an output that includes the sum and difference frequencies of the two inputs. The sum frequency is even higher than the original signal, thus it is useless and is filtered out. The resulting signal is a single frequency (plus harmonics) in the audio range.
The frequency of the volume oscillator is used to control how much the audio signal is amplified. As you move your hand, the frequency changes, so the amplifier's gain changes, and thus the output volume changes.
Neat problem.
I think you'll need the GSM repeater. Considering they are not that expensive compared to the cost of other equipment you'll need, it's probably your best bet.
The geology is going to have some impact. If you happen to be surrounded by a lot of metal, you're in a big Faraday cage. If you happen to be surrounded by radioactive material, that will interfere. (Many stone buildings give off significant amounts of radiation.) It's quite possible that you will run into some strange problems. If getting it right quickly was a concern, I'd hire a professional to bring in test equipment and determine what's feasible.
Best Answer
The booster comprises a high-Q tuned circuit in which the coil has a large loop area.
The large area allows it to capture more signal power than the radio's built-in antenna can, and the fact that it resonates with the signal allows a large current to build up in the coil.
Since the radio is primarily sensitive to the magnetic component of the radio signal anyway, it couples readily to the current flowing in the booster, producing a stronger output than it can on its own.
Also, the high selectivity of the booster helps to eliminate adjacent channel interference, further improving the perceived quality of the signal.