Why electron energy increases in an excited state? After absorbing a photon with the minimum threshold frequency an electron gains energy and enters an excited state, but according to Couloumb's law a greater distance between two charges will result in a smaller energy, so does this mean a small amount of energy is lost to still result in a net gain in energy? 
 A: 
but according to Couloumb's law a greater distance between two charges will result in a smaller energy

should read  
“but according to Couloumb's law a greater distance between two charges will result in a smaller force of attraction”  as Coulomb’s law is to do with forces and not energy.
The electron is not held in the atom as strongly as before the photon had been absorbed.   
An electron in the ground state of a hydrogen atom has an energy of $-13.6\, \rm eV$ if one defines the energy of an electron at infinity as $0\, \rm eV$.
If a photon of energy $10.2\, \rm eV$ is absorbed by a hydrogen atom in its ground state the electron is promoted to the  $-3.4\, \rm eV$ energy level which means that in this state the electron has more energy (the energy is less negative) than it had in the ground state.
A: When an electron is bound to the nucleus, it exist at a certain energy level around the nucleus as per QM.
it is stable in this energy level. It is because:


*

*kinetic energy keeps it away from the nucleus.

*EM attraction keeps it close to the nucleus.

*As soon as the electron would move closer to the nucleus, the Heisenberg uncertainty principle will cause the electron to gain momentum (kinetic energy), and that keeps it away from the nucleus.
The three forces equal out, and the electron is stable at that certain energy level as per QM.
Now when an electron absorbs a photon, it gains kinetic energy from the photon. The photon does not carry EM charge. Because of that, the electron does not gain EM charge from the absorbed photon itself. Only the kinetic energy of the electron will rise. The electron will move further away from the nucleus because of this kinetic energy (from the photon). This energy level for the electron is not stable. It will move down to a lower energy level if available, and emit a photon. By emitting a photon, the electron will lose kinetic energy, so it will move closer to the nucleus. It will return to the ground level.
You are right, that when the electron absorbs the photon, it will move further from the nucleus, and the EM attraction between the nucleus and the electron will be smaller (then at energy levels closer to the nucleus) at this energy level that is further from the nucleus. Because of that, the electron will be further from the nucleus and thus the Heisenberg uncertainty principle will not be so dominant. The electron's position will be known with less certainty, so its momentum will decrease (this effect of the Heisenberg uncertainty principle will add less to the momentum). 
So at the energy level away from the nucleus (compared to the energy level closer):


*

*kinetic energy will be more because it gained it from the photon

*EM attraction will be less 

*Heisenberg uncertainty principle will add less to the momentum
These three will equal out, and the electron will be at this energy level. This energy level will be less stable then the ground state. The electron will only be at this level until the lower energy level will be available, when the electron moves back to the ground state.
