This is a very deep subject so we will give you the short version!

“What aspect of mouthpiece design mostly effects tonal signature? I'm referring to the strength of various overtones with regard to timbre quality, with a goal toward producing a sweeter (richer in high overtones?) tone.”

The most important thing in timbre quality is the players lips, to include lip oscillations and lip equilibrium (depth of average lip penetration). This is your acoustical fingerprint and everyone will be different. Match between player, horn, and mouthpiece is also important. The lips must be free to vibrate to attain your acoustical fingerprint and well as everything in the GR 3 Rules of Brass playing.

“with a goal toward producing a sweeter (richer in high overtones?) tone.”

Many would argue that a sweeter tone has less high overtones. This is a preference that is defined by each player. To enhance upper overtones the pop pitch frequency of the mouthpiece must be raised. Therefore, less volume and it can be achieved in several ways and here are 2 of them: reducing the mouthpiece volume or increasing the lip engagement (this decreases volume too). This will vary with each different set of lips.

“Do mouthpiece makers take into consideration "popping frequencies"?”

At GR Technologies, we list all the cup volumes in cubic inches. We do know the pop pitch frequency for every GR mouthpiece. We also know this will change when you put a set of lips into the mouthpiece. Again this is player specific. Pop pitch frequencies for average mouthpieces will vary from about 550 hz to 800 hz. We did some testing with Thomas Moore PhD @ Rollins College, also ITG technical editor. This info may still be published on his Rollins College physics dept website. Lip engagement can raise the Pop pitch frequency as much as 150 hz or even more for certain players. This is called dynamic testing. In the past all the physicists did static testing of a mouthpiece so results will not match a person. As the pop pitch frequency raises so does the amplitude of the upper partials.

“What effect has relative length of parallel part of the throat on timbre and intonation? Remedy for airy-ness (detectable hiss) of tone?”

Again just looking at one parameter will not give you an answer. The entire system must be understood and considered. Bottom line with airy-ness the lips are not vibrating efficiently and excess air is passing with each vibration. Cause could be impingement, rim, pressure, face, lip equilibrium, correct backpressure, gap, venture, leadpipe, bell, any aspect of the mouthpiece, a sever mathematical discontinuity in the system. This is why we created the GR Playing tests to diagnose and correct problem using an organized system. You can make up your own system as long as it is organized. My first guess without testing is the system is too open; lips are impinged or not supple to vibrate. How soft can you do a breath attack? If this is hard you have other problems.

“I know there are many other mitigating factors associated with the instrument itself (receiver gap, leadpipe, bell etc.) but I'm just interested in mouthpiece acoustics right now. “

If you only look at the mouthpiece and not the entire system (player and horn) you will only chase your tail. It the match and will differ for everyone. See GR’s Rules of Brass playing #3.