Q1. "I've read comments to the effect that "as you approach the state of the art, solid state and tube equipment begins to sound very similar".
I would agree with that, however then hasten to add that "as you approach the state of the art” smaller differences have a greater effect. An analogy is wet sanding then polishing a high gloss finish on a speaker or car, the more one polishes the better all the details (and alas defects) are revealed. Thus a paradox, they sound more alike but the perfectionist builders and owners of high-end hi-fi care ever more about the ever smaller differences. Also keep in mind there is a niche branch of hi-fi tube electronics designed for a very specific house sound; beauty first, accuracy second. The sound of a Shindo Masseto Tube Preamplifier will never converge with a Boulder 3010’s.
Q2. “I've read that tube equipment tends to generate more even-ordered harmonic distortion. Does this mean that distortion in a tube preamp can cause the sound to come across as "richer" and "fatter"?”
Yes, and a universally recognized master of solid state hi-fi design Nelson Pass has shared his insight that the phase of the distortion harmonics vs. signal also has an effect.
Q3. “I've read with interest Roger Sanders excellent white paper on tubes vs. solid state, and his comments make sense, for power amps. But what about preamps? Do preamps clip?”
I would say in 99.9% of the use cases that hi-fi stereo preamps do not clip. Typical power amps only require 0.5 to 2 volts of signal input to output their hard clipping maximum power and preamps can supply 10 times that.
Q4. “It seems that several factors greatly influence the sound any preamp - the quality of the power supply (bigger, stiffer and isolated in a separate box seem to be important), the quality of the volume control, the quality of the transformers...Frankly I'm out of my depth here, but some of you might be able to help me understand this better.”
That is not so much a question as an outline for a master thesis on hi-fi electronics design. I would agree with your list adding quality of all parts to it while noting that personally I have come to doubt isolating the power supply in a separate box is always necessary. Back to the tube vs solid state question one of my pet theories is that a good chunk of the seemingly intrinsic difference flows from the fact they operate in such different voltage domains. The passive components (resistors, capacitors, diodes, switch contacts, etc.) in the tube circuit have 100s of volts on then compared to the solid state circuit with 10s of volts. Thus the ratios of the signal voltage to the parts’ non-ideal secondary characteristics are very different in the two types of circuits.
Q5. “Some highly rated tube preamps have multiple tubes, and use some in the power supply. Others have very few. How should a person think about this?”
Engineering design takes a design requirement in this case ‘sounds amazingly like real music, interfaces with my existing equipment, is reliable, X inputs, to Y outputs in format(s) Z, user interface like this all achieved in a cost budget of so-much’. Then one considers every circuit ever devised (and newly invented if one is feeling especially confident this week) these weighted towards what has worked for the designer(s) in previous designs. Fitting all that together is very much an art of weighing the compromises of the various options and picking the tradeoffs that best deliver the desired results. These tradeoffs’ weighting also very much include the designer’s and brand’s overall philosophy. As an example I highly value ‘transparency to the source’ and that tends to be easier (and less expensive and potentially more reliable) to achieve with fewer tubes as few as one per channel for a line stage. HOWEVER, if the design requirement also includes the ability to drive low impedance loads like 600 ohm pro-audio gear or (quite likely in 2019) headphones that calls for more tubes with some providing voltage gain and others following later in the circuit providing current drive.
Naimnut observed “use some (tubes) in the power supply” and Mark Korda asked “does vacuum tube rectification vs. silicone diodes for rectification have anything to do with the sound?”
First short answer is across all types power supplies have everything to do with the sound. Sometimes we find tubes in the power supplies simply because the designer(s) and/or the brand and/or their customers are dogmatic ALL TUBES, NO SOLID STATE. Sometimes because they look cool, I am thinking the blue glow of mercury vapor rectifiers. Technically using a tube in a place like the pass element of a regulator can be justified because it is regulating the 100s of volts it was designed for and in such an application if a fault develops a 6550 will survive while a MOSFET dies immediately. In rectification when compared to silicon diodes tube rectifiers generate less high frequency noise when they turn on 50 or 60 times a second. The solid state designer can answer that with filters around the silicon diodes or by using Schottky diodes. The tube rectifier can be quieter, but it will likely not deliver as much current as quickly as a solid state design.
The above just touches the questions asked above from a decidedly engineering perspective. To rework the old saying about real estate what counts is ‘implementation, implementation, implementation’. As you consider your upgrade it is fun to count the tubes like a watch collector counts the complications in a mechanical watch movement. I council to consider that as part of the story while also weighing the capability of those who take a clever design and turn it into the actual thing you unbox and wire into your hi-fi rig.