How does this trapped sweat evaporate?
With a perfect VBL, it doesn't.
Say I am wearing a thin sock, a VBL sock, an outer thicker sock, and my boots. The VBL keeps the sweat between the thin sock and the VBL, but isn't my thin sock totally drenched?
Since it's a boot, how does one dump heat?
You still lose it by all the normal methods of heat transfer, that is, convection, conduction and radiation. The only difference is that evaporation doesn't amplify convection or conduction. In convection, the air moves around allowing more sweat to evaporate (which loses heat) instead of becoming saturated. In conduction, your wet socks would allow heat to flow directly into the cooler surfaces of your shoe. A VBL doesn't necessarily stop convection, so you can still have fresh air flowing into your thick socks. Since your socks are not perfect non conductors, you'll still have some conduction going on.
Is the idea that the baselayer might be sopping wet but the outer layers are not and so I am still warm?
You got the idea.
Does one need to ever dry a VBL? Does a VBL stink?
If it's wet, I would want to dry it before putting it back into my pack. If it's below freezing, drying it may be as simple as laying it out for a few minutes and then shaking the ice out of it. Stink? Every piece of fabric that touches me stinks...
If you sweat while sleeping without a VBL and wake up steaming, wouldn't you sweat even more with a VBL on under your bag? So, you're uncomfortable and wet but your bag is dry?
This takes getting used to. Ideally you'll find the perfect temperature at which your sweat is at equilibrium. I always let myself get too hot, so I sweat all night. I usually find the sweat pooled up under my pad if I'm using a bivy. Once it's under my pad it won't evaporate again, so it's no longer a problem until the morning when I need to shake out the sweat.