Flow from higher potential region to lower potential region In Thermodynamics,  it is said that heat transfer takes place from higher temperature to lower temperature. Things having higher temperature have higher potential energy because of more vibratory motion and other kind of motion of atoms and things at lower temperature have lower potential energy.
But in electrodynamics, the electrons move from negative terminal of the battery which is at lower potential to the positive terminal of the battery which is at higher potential. 
One fundamental law of physics is that things go from higher potential region to lower potential region. Why does they do so? Why does heat flow from higher temperature to lower temperature and not vice versa?
Second question is why this thing is not followed in case of electrons? Am I missing something or the concept of potential is also relative, like for electron,  negative terminal is at higher potential and positive is at lower.
 A: 
But in electrodynamics, the electrons move from negative terminal of the battery which is at lower potential to the positive terminal of the battery which is at higher potential.

This entirely due to a naming convention. Had we decided to call the charge of the electron positive instead of negative the electrons would have moved from the positive (higher) to the negative (lower) potential.
The potential energy at the negative electrode arises from mutual electrostatic repulsion between charges with equal sign (electrons). At the negative electrode there are more electrons (particles of the same sign charge) and thus higher potential energy than at the positive one and this causes them to flow from the 'negative' to the 'positive' terminal.

Why does heat flow from higher temperature to lower temperature and not vice versa?

You more or less answered that question in your own first paragraph although the higher temperature causes the molecules of the hotter material to have more kinetic energy, not potential energy. When heat flows from hot to cold, molecular kinetic energy is transferred from the hotter to the colder area. This can obviously not happen in the reverse direction.
A: Gert's answer states that heat cannot flow from lower temperature to higher temperature because it is obvious. I suggest that your first question could be answered with the concept of entropy. To put it simpler, particles and their energy levels would tend to a more probable state.
For example, when you throw 6 dices, the probability of it getting all six sixes is far lower than a somewhat random distribution e.g. {1,2,3,4,5,6}. Hence given some time and a chance to bump around, eventually the heat would be dispersed because it is more probable.
Therefore entropy must increase (Second Law of Thermodynamics) (and/or the decrease of entropy would lead to an overall increase of which, in the case of heat pumps) and heat transfer from hotter part to colder part satisfy this condition.
For the second question, as Gert has stated it is purely the problem of nomenclature.
