So moving to a smaller process to shrink the borders does not affect the amount of light captured for each pixel because of the gapless microlenses?
Pretty much, yes.
If microlens size = photo site + border, then it would seem that a larger pixel-with-microlens would gather more light than a smaller pixel-with-microlens. Are you saying that the resolution (given the same sensor dimensions) is higher for the smaller pixels so when you compress the image to the same resolution as the sensor with the larger (fewer) pixels, the overall light/data collected for the multiple smaller pixels, now sized-down to the lower resolution end up producing essentially the same image quality?
Yes, though if you do the down-sizing properly, the smaller pixels will generally win, and quite easily.
Am I understanding this right? Does this mean that if I want to enjoy the same image quality as the sensor with fewer pixels I have to compress the resolution of my images to match?
That depends on what you mean by image quality. Resolution? Noise? With the smaller pixels, you have the option to reduce noise at the expense of resolution. On the larger pixels, that part has been done for you and you have no choice.
One other thought: microlenses perfectly focusing the light on the photo site sounds great on paper. How precisely do the lenses do this in the real world? If they're nearly perfect, how in the world do they accomplish such precision on such a small scale? Simply amazing to me...
The efficiency varies with the design, but it's quite close to all of the light. They do this using the techniques of photolithography, which is quite a precise thing, especially in the more modern versions.
If the microlenses do their job, then I guess it's not light/surface-area that makes the difference between crop and full frame. Could it be that for the smaller pixels, there's more opportunity for noise to be introduced by the supporting circuitry? Something must be happening, because it seems that sensors with larger pixels seem to do better for noise at high ISO.
They do this because they use more sensor area, not because they use larger pixels. When you have the same f-stop, the light intensity (called "illuminance" - light per unit of area) is the same (for a given scene), and that means a sensor with more area captures more light. Since signal-to-noise ratio goes with sqrt(total light captured), more area (bigger sensor) means better signal to noise ratio for the same f-stop. That's why larger sensor perform better in low light.
Another way to look at the same thing is to express f-stop as its definition - focal length / aperture. So, a lens with a 100mm focal length and a 25mm aperture has an f-stop of 4 (it's often written as its reciprocal - 1/4 or 1:4).
Well, let's say you want to use your 100/4 on your full-frame camera. To what do you compare? Well, on a 1.6-crop camera, you might use the same lens zoomed out to 62.5mm so that you have the same angle of view. 62.5mm / 4 = 15.625mm compared with 25mm on the full-frame camera. That's a lot smaller hole for the light to squeeze through, and so you get a lot less.
The two explanations are equivalent.