OK, let me try to answer from the top. But first, a little background. I play Adams now, played a Sterling for about 20 years before that, and played a Besson for about 20 years before that. I don't jump from horn-to-horn without good reason and careful comparisons. My primary goal in choosing a horn is getting a rich, singing sound. If you don't like a horn's sound but like everything else, it can be tough to change the sound. I stayed with Besson for so long (during the dreaded mile-high-6th-partial years) because I like its sound better than anything else, even though (as I acknowledged at the time) other horns had better response and intonation. I changed to Sterling because it had the character of sound I wanted and offered a little better response and intonation (both of which got even better over the years). I changed to Adams because it had even better sound, the best intonation of any horn I've tried, and the best response of any horn I had tried.
I was very used to having a trigger on my Sterling and liked the relaxed way I could play by using it. The Adams' intonation is good enough that I chose to not buy the trigger option. (Which I thought to be a good thing, because trigger adds weight and vulnerability to damage.) I started with a .55 horn with a SS bell. Then later I moved to a .60 horn with a SS bell. That is what I use today.
Now to specifics:
1. How does red brass, Gold brass, regular brass, and sterling silver affect the sound?
Personal taste enters in here, even in the way we describe the differences. If I were going to advise on the safest choice, I'd say go with the sterling silver bell. But I also like the red brass a lot myself. To me, the red gives a more lively sound compared to yellow (and SS for that matter).
2. How does the thickness of the body of the horn affect the sound?
I did not hear it as affecting the sound so much as it affects the volume ability of the instrument and the response. The lighter horn responds more easily but can't project quite as well. The heavier horn does not have quite as quick a response but projects better. The heavier metal can handle your playing better if you tend to use a lot of air/power. If you are by nature a lighter player, you might enjoy the lighter metal's feel.
3. How does heaver valve caps affect the sound?
I can't swear that I can tell a difference. I use the standard caps on 1-2-3 and a heavy cap on 4. Sometimes I think the heavy cap makes a difference and sometimes I don't.
4. Does the bell size matter?
Yes it does, as far as I can tell. Generally a larger bell gives you a larger sound, but has less focus. And vice versa. I came to this conclusion during my Sterling Virtuoso years, when I had ample chance to try identical (as much as they can be) horns but with different bell diameters. For large ensemble playing a large bell is nice. For chamber playing and recitals it might get in your way a bit.
5. How does the custom receiver affect the sound?
Subtly. It makes a difference to my ears, but it's hard to describe it, and it is small. The much more significant factor is the way it affects the response and slotting. See the article linked above in Rick's post for more insight.
6. How does Tuning Trigger affect the sound?
Probably a subtle difference in sound. As stated already, Adams prefers horns with less bracing so they are freer to vibrate. That theory seems to work. Certainly it has been generally found/believed over many hears that reducing bracing helps with response. At the very least it has an effect of some kind. Having a trigger means extra bracing, which should mean less vibration. Matt van Emmerick believes as I do, that the horn does not really need a trigger for intonation. But he likes it to be able to perform certain effects and for muted passages. In my case, there are times when I'd like one now and then, but it's not too tough to do without it. And my Sterling suffered damage to the trigger on a plane flight to ITEC 2010 that would not have been troublesome on a non-triggered horn, so you have a little risk involved. There is also a little extra maintenance. My advice is to try the horn. If the dealer has only a triggered model, see if you can play it easily without the trigger. I confess I'm kind of a wimp about lipping notes (on my Besson I used alternate fingerings instead), so if I can get by without a trigger, then most players probably can. That's especially true when I say it at this point in my life, when my practice schedule is hardly worth calling a schedule and my chops are not in the shape they were when I was playing for a living.
As said above, a .55 or .60 is a good place to be. I spent a great deal of time in Europe last year playing all the possibilities. The .50 was really fun, but a touch to light for my overall use. (Actually, one of the sweetest Adams I played in Austria last year was a .50 with red brass bell. Lovely horn, but maybe not the best choice for playing in a large band.) The .70 and .80 had impressive dark sounds, but seemed to heavy in feel and sound to me. Most pros use the .55 and .60 with SS bell. Matt Tropman uses a .50 with SS and a couple other really fine players I know prefer the heavier metal.
With all those options, it would be great to try them all, but that's darned hard in the USA (actually, it's impossible as far as I know). If you have some extra money, you could have fun and make a trip to the factory. Get a cheap flight to Amsterdam, train to Adams, and they will put you up in a hotel for the night. It would add 15-20% or so to the overall cost, but you'd really be able to get picky! (I should have warned you that I love to spend other people's money
)
See forum 
