Challenge 1 Solution
We should expect such materials to have high specific heats. Here is why: when we add a fixed amount of energy to such a molecule (i.e. one that can vibrate or rotate internally) some of the energy we add goes into increasing the internal vibration and/or the internal rotation. This means that less of the energy is available to increase the kinetic energy of the molecule as a whole.
Challenge 2 Solution
When the cold water is added to the hot water, a large part of the energy transferred from the hot water to the cold water will go into the internal motions of the water molecules while some of the energy transferred from the hot water will go into increasing the average kinetic energy of the cold water molecules.
In the other container, when the cold aluminum cube is put into the hot water less of the energy transferred from the hot water to the aluminum goes into the internal motions of the aluminum molecules and thus more is available to increase the average kinetic energy of the aluminum molecules.
If you wanted the container with the aluminum to reach its equilibrium temperature as slowly as possible, the aluminum should be in the shape of a sphere – why?