It seems like among the electrical conductors there's a relationship between the ability to conduct heat as well as electricity. Eg: Copper is better than aluminum at conducting both electricity and heat, and silver is better yet at both. Is the reason for this known? Are there materials that are good at conducting electricity, but lousy at conducting heat?
Material Science – Why Materials Better at Conducting Electricity are Also Better at Conducting Heat
electricityheatmaterial-science
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In order to have the hot side remain hot, and the cold remain cold, while transmitting the electricity, ideally a substance is desired that conducts electricity well, but heat poorly.
The two ends are assumed to be connected to heat sources or sinks that maintain a constant temperature at those nodes. Hot electrons and cold electrons travel through the material at different rates, leading to a net flow of electric charge towards one end.
https://en.wikipedia.org/wiki/Seebeck_coefficient#Charge_carrier_diffusion
- How do you make a series circuit with lots of thermocouples? Surely this must be done, yet when I draw a diagram, I can't come up with a way that doesn't look like two identical thermocouples are simply operating in opposing directions.
A series connection of thermocouples is called a "thermopile". It's connected like this: 
- Suppose you take two materials with an electropotential: Cobalt and Lithium. If you could make two junctions between them, one a cold, and one a hot, could you take a third material (copper) and use it to connect the two junctions? Does that ruin the thermoelectric effect?
Not sure what you're asking. You want to connect wires made of lithium, cobalt, and copper in parallel?
Electricity needs charged particles (or quasi-particles) to conduct it. Heat can be conducted with almost any quasi-particle. Diamond is one of the best conductors of heat in existence, and it's because of phonons, ie quasi-particles of lattice vibrations, which are strong because the diamond lattice is strong.
Best Answer
See http://en.wikipedia.org/wiki/Thermal_conductivity In metals, I think it generally has to do with the higher valence band electron mobility, but it gets more interesting elsewhere.