Some physics: Does thicker or thinner conductive material transfer more heat? and more...

A thicker piece of material spreads heat more evenly, and a thinner one will heat up and cool down faster (less thermal mass), but which can transfer heat faster? I believe the theory is that a thicker sheet of metal will take longer to heat up, but once steady state is achieved, can actually move more heat per unit time. Is that correct? Or is heat transfer the same regardless of thermal mass? Is this property different longitudinally or transversely, i.e. with a sheet (like pan) or a wire? Can thicker gauge wires carry more current, or simply carry current further with less loss, or carry it faster? To the last question, does electricity moves at a constant speed in a given material regardless of shape/size? Does it ever approach the speed of light? Is there a coefficient for it's speed for any given material (is that essentially the same as electrical conductivity x density)?

Some of these are more quantum questions, a little off topic, but interesting none the less, and I think we can generally conceptualize about heat and electricity in similar ways.

But how about heat, specifically? Does heat radiate at the speed of light in a vacuum? How about in (still) air? More to the point, for this forum, does conductive heat move at different speeds in different metals (I would think so...that's kind of the definition of thermal conductivity, right?), and is its speed or direction of spread influenced by the size/shape/thickness of the metal? When you apply heat to one end of a thick wire, will it propagate first to the edge of the wire near the heat source or first move longitudinally into the wire (assuming it is one solid piece of metal, and not braided)? What would the calculations look like to determine what temp the outer edge of a wire would be by the time the opposite end of that wire x meters long by x meters in diameter increases its temp by x degrees with x units of heat input at other end?