[Coda]

I mean that when you're dealing with chemical batteries or solar cells, you're getting a bunch of free electrons on one end of the circuit, and since like charges repel, they get pushed out as soon as there's a place where they can actually go. And they're not specifically looking for a positively-charged place to go; they're just looking for a less-negative place to go (because being less negative means it's going to push the electrons away less), and the Earth is so massive that you can add a whole lot of electrons to it without making a significant difference to its net charge, so it stays overall pretty neutral no matter how much you pump into it.

You CAN do it the other way. You CAN have a reaction that binds up electrons and leaves a positively-charged void that's desperately trying to attract electrons. That's a negative electric potential. It'll happily draw those electrons from the ground (because the Earth has so many electrons that it'll never miss a few) if there's no place better to get them from.

Chemical batteries do this at the positive terminal, so given the choice to travel from the negative terminal to the ground or to the positive terminal, the positive terminal is twice as attractive of a destination. Solar cells are comparatively weak because the only force pulling electrons through the circuit load is the fact that, after a photon forces an electron to start moving down the wire, there's not an electron there anymore so there's room for another one to come in from the other side.

You're not too far off when it comes to why colder metals conduct better. A hotter material has more random motion in it, the atoms in it are moving faster relative to each other, so electrons have to be moving faster in order to make the jump from atom to atom. A colder material keeps the atoms closer together.

I don't completely understand the thermal conductivity thing myself. I know it because i read it. However, no, it doesn't mean that a hot metal pot will get hotter more easily. It means that the heat will spread out more evenly once it's gotten warmed up compared to at the beginning. Instead of a pot, think about a long metal pipe with a heat source at constant temperature at one end. If you draw a graph of the temperature of the other end of the pipe, then it'll be a curve going upward instead of a straight line -- the hotter the far end already is, the more rapidly it approaches the temperature of the heat source.

If I had to take a guess as to why this is true: Imagine a billiards table. In a cold material, the balls are all close together, with some space between them. If you shoot the cue ball into the cluster of balls, the energy is going to get spread out pretty fast because each ball will bump into more balls as they start moving. But if the balls are more spread out, then when you shoot the cue ball, whatever it hits is going to immediately go zipping off because there's not as much in its way, and so it can transfer its energy to a place farther away on the table sooner.