1.) They sum up both pitch oscillators' signals for additive mixing over the demodulator diode D4. Since the latter has an exponential U/I curve, the result is a multiplication of both signals. e^(a+b) = e^a * e^b
2.) When nothing is in proximity of the volume loop, the series resonant circuit formed by the static loop capacitance and L7 to L10 is in resonance with the volume oscillator. Thus a resonant current flows through L7 to L10. The current which flows through L7 makes that a voltage proportional to this current will appear between both ends. This voltage is rectified by D1, C12 acts as a load capacitor. A simple, classic one-way rectifier...
3.) D4 acts as mixer/demodulator diode (see my answer to question n° 1). R23 and R24 form together the load for the demodulated signal and a voltage divider to adapt the signal level to the input of U-3b. At the same time, R23, R24 and C23 form a low pass filter to eliminate remaining RF from the demodulated signal.
4.) As I stated in one of my above postings, Hammond 153x series or Miller 630x series, values as in the above schematics. Some more recent industrially produced Etherwave Theremins use only three coils on each side. This requires a much more sophisticated circuit board layout with less parasitic capacitance. When building the circuit "by hand", the use of the original 4 coils on each side is strongly recommended for stable operation and easy tuning. Once everything works perfectly, you may still experiment and short L10 and L4.
5.) Big electrolytic capacitors are good for buffering DC but they have very bad RF properties (high parasitic series inductance). Thus C29 and C30 (ceramic or tantalum capacitors, not electrolytic!) are added to form a virtual ground path for the RF and preventing the integrated voltage regulators from oscillating themselves, causing interference or heating up.
6.) The third schematic with C10 and C11 = 100uF and C11 to ground is much more stable. The older variant with 0.47u or 1u capacitors and C11 to +12V was less stable and had the tendency to make the reactance tuning stage around Q5 oscillate itself at around 330Hz under some conditions.
In general, you should understand that all RF components in a theremin are much more critical than in a radio or tv set. Thus you should always use the indicated and approved components and not try to do wild replacements by parts with "almost similar" values and properties, because the latter will almost always give unstable operation and/or unpredictable results.