1) Although the CWA and HX may look very similar, their function is quite different.
Both have primary connected to the mains, secondary connected between tap (or the 120V top end) of the primary and the lamp and a magnetic shunt which forms an extra inductance seen as in series with the output.
But the difference is how this inductance behaves and how it is used:
With HX this inductance forms the main ballasting impedance, so an element essentially controlling the arc current. With this arrangement, the magnetic shunt is designed in a way it does not saturate. When the mains voltage increases, it leads to higher drop across the ballasting reactance, so higher arc current.
With CWA the main ballasting impedance is the capacitor in series with the lamp. The reactance of the leakage inductance (controlled by the shunt) is then subtracted from the capacitor reactance. Here the shunt is designed so, it saturates at the rated lamp arc current. This then gives the ballast interesting property: If the arc current becomes higher than required (e.g. because the mains became higher than nominal), the shunt is driven deeper into saturation. This then means the leakage inductance gets reduced. Because this is in fact subtractied from the capacitor reactance, the total ballasting impedance becomes higher, so going against the increase of the arc current.
In other words this CWA system is able to keep the arc current in tighter tolerance even with rather wide mains voltage variation, so (assuming constant lamp arc voltage) leading to constant power delivered to the lamp even when the mains voltage varies. And from this it got its commercial name (Constant Wattage), although in reality it means mainly constant output current.
2) HX is way simpler, so cheaper. And also it does not need the capacitor, which is rather sensitive component (its lifetime is way shorter than the transformer itself, plus is way more sensitive to high operating temperatures). Because the lower wattage MV's are most often used mainly in areas with less requirements, those use to be more often equipped with HX.
Also HX has the arc current drooping with the arc voltage rising up, which improves thermal stability mainly with saturated vapor style lamps, mainly important with lamps which performance (the cpolor,...) sensitive to the arctube temperature like pulse start and ceramic MH, so these use to use HX ballasts as well.
On the other hand the high power MV or HPS use to be used in installations with longer wiring, so worse mains voltage stability, so benefit way more from the lower sensitivity to the mains voltage variations, hence the CWA is being preffered. As these are burning way higher power, the higher ballast cost is not that visible, the tighter power operation offsets the extra cost of the CWA.
3) See 1). It is not the capacitor, but the series combination of the capacitor and a saturable inductance, what is responsible for the regulation.
4) The capacitive nature means the ballast has lower impedance at higher harmonics, so the current tends to be more spikey (higher crest factor). The saturating inductor is making this even worse. What is causes: The power is dictated by the area under the current half wave, but the electrode stress is more related to the peak value.
The HX, because mainly inductor, yields very close to sinewave current shape, so the peak current is Pi/2 = 1.57 times the average. So if the lamp has 110V drop and has to get 175W, it leads to 1.62A average, so 1.8Arms, so about 2.54A peak current.
The CWA forms rather high spikes, so for the same 1.62A average it forms peak currents of 3.2A or even higher, so way greater stress for the electrodes. Now during these peaks the emission will become very marginal, so the cathode drop increases, so does the sputtering by faster ions. But this is to some extend a matter of electrode design - so commercial lamps rated for CWA are designed to handle this while still maintaining their rated life. And when the mains is really fluctuating a lot (like long stretches of road lighting), the tighter arc current control (preventing an overcurrent when the mains becomes higher) actually compensates for the spikey current effects, mainly for lamps rated for CWA operation (MV, higher power HPS, probe-MH).
5) Yes. The primary along with the resonant center coil stabilize (regulate) the voltage at the center coil (controlled by core saturation within that center coil), the shunt between them allows the voltages to differ between the input and the resonant coil.
Then the secondary vs the resonant center coil forms just a basic HX transformer ballast - so forms an inductive ballast for the lamp.
This ballast may seem as the best - it regulates the mains, as well as it stabilizes the arc (higher arc voltage leads to lower ballast output current), plus the current is without the CWA spiking.
But it is at rather great expense: Three coils, with not much possibility to share even sections of the winding means 3 windings with their power losses, copper cost size. The cost and size must then be even bigger, if you want to bring the losses back. And that means very high cost and size and rather high losses (for the cost and size).
But in the past this was worth "to pay" with the high power and high performance lamps (which use to be more demanding for temperature stability, as well as the power tolerance and crest factor), their higher performance tends to offset the extra losses.
But in the modern times this has been surpassed by the electronic ballasts.
6) Probe vs pulse use to be designed for different electrical characteristics (different currents, ballast OCV even when not counting the HV ignition pulses), one of the main differences is the pulse lamps require the ballast to reduce the output current when the lamp arc voltage becomes higher (so a HX). On constant current CWA the lamps become very thermally unstable (there is a positive thermal feedback with saturated vapor pulse start lamps).
7) On Probe-start MH ballast it can work directly, it was actually an EPA directive somewhere from 2009 or so, which banned ballast makers from rating MH ballasts also for MV officially, before that all probe start MH were even officially rated for the same wattage MV. In fact the MH were originally targeted to work on MV ballasts, but later was found out the MV OCV is not enough to start them reliably, so MH ballasts were needed with higher OCV. But those were rated to operate the MV as well.
For choke ballast the main voltage must be at least 2x the arc voltage. So in Europe (and other 230V areas) practically all HID's (MV, HPS, MH; maybe except the most recent low power MH) are designed to work exactly on that, a series choke ballast (eventually with an ignitor). But for 120V feed that would mean arc voltage to be maximum 60V. And (except the really low wattahe HPS) that would lead to lamps with very high relative electrode losses, so low efficacy. Electrodes have about 15V drop, so with 60V arc that means the full 25% of power is turned just into electrode heat.
With low power HPS lamps this could be still acceptable, as too thin (too low current) arc also means extra losses, so the 60V arc is not that far from the optimum, so the low cost (so allowing some extra budget to reduce losses further) and low losses of a simple series choke the overall system efficacy is practically optimal.
But with other lamp types and higher powers the optimum is so far in the higher arc voltages, even the higher losses of an autotransformer ballast mean the total efficacy to remain significantly higher than a 60V arc design would offer.
