The second technique you mention is the way to do it, using what is called a "Software Defined Radio" or SDR. Many radio amateurs are using SDRs, and the simple ones are very cheap, about 30 dollars for a kit that down-converts the input into in-phase and quadrature baseband audio output which is fed into the stereo inputs of a PC sound card for digital signal processing. However, they are using relatively low-frequency signals on the HF amateur radio bands, and the hardware doesn't use any exotic components. Digitising VHF signals as you require and receiving several channels simultaneously is going to be rather expensive, the ADC alone is going to cost about 50 dollars and you will also need an FPGA and a DSP, unless you convert down to baseband and do the DSP on a PC. You will need a lot of high-frequency design experience, be able to develop code for the FPGA, write DSP code and be able to design a high-speed multilayer PCB, so you should start studying. :)
As for cost, I'd estimate 500 dollars for the hardware, including the PCB, assuming you designed it yourself.
Linear Technology makes suitable ADCs that can downsample at 750 MHz! They were good enough to give me a couple as free samples. I have suitable FPGA and DSP boards, so it's just a question of putting them together. :)
The complete derivation from Maxwell's equations fills entire college-level textbooks, and is too involved to get into here.
But when considering radiation from an antenna (a current flowing in a linear conductor), it boils down to the fact that there are several distinct components to both the E (electric) and H (magnetic) fields around the antenna. For the H field, there is one component that is proportional to 1/r2 and another that is proportional to 1/r. For the E field, there are three: a 1/r3 component, a 1/r2 component and a 1/r component.
The 1/r3 term is the dipole electrostatic field, which represents the energy stored in a capacitive field. Similarly, the 1/r2 term represents the energy stored in an inductive field. This represents the "self inductance" of the antenna conductor, in which the magnetic field produced by the current induces a "back EMF" on the conductor itself. Only the 1/r term represents energy that is actually carried away from the antenna.
Near the antenna, where the 1/r3 and 1/r2 components dominate, the phase relationship between E and H is complicated, and these fields do indeed store energy in the manner that Olin describes, and return energy back to the antenna itself.
However, in the "far field" (e.g., more than 10 wavelengths away from the antenna), the 1/r components of the fields dominate, creating the propogating electromagnetic plane wave, and these components are indeed in phase with each other.
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Your mention of the frequency (97.5 MHz) tells us this is an FM receiver. (AM will behave differently, as will other modulation schemes).
Because FM is encoded by modulating the signal frequency, anything to do with AM is undesirable. To deal with this, most receivers over-amplify the signal until it becomes larger than the later stages can pass. The signal then "clips" to the voltage of that amplifier. This stage is called a "limiter"--it limits the amplitude to some fixed value. In theory, any signal weaker than that drops out and just becomes noise, and any signal stronger than that has a very nice fixed level that the FM detector can handle without having to worry about amplitude variations.
The amplifier-limiter stages create a phenomenon called "capture", where the strong signal tends to eliminate the weaker one. This is why you hear only one station.
If the signals were very close in strength, you would indeed hear them "mixing together", but that only happens for a fraction of a second as the signal levels rapidly change (presumably, you are in a vehicle), so you normally don't hear that.