Energy of Electrons Why the electrons farther from the nucleus have more energy than the electrons nearer to the nucleus ? What factors make the farther electrons to possess more (potential) energy than that possessed by the electrons nearer to the nucleus ?
 A: Electrons closer to the nucleus will have a lower, meaning more negative, potential energy. Electrons further away from the nucleus will have smaller valued negative potential energy (absolute value is smaller), meaning electrons farther from the nucleus have greater energy. It might be counter-intuitive, but it makes sense when you realise that it takes more energy to remove an inner-shell electron and less energy to remove an outer-shell electron.
Mathematically,
$$E  = - \frac{Z^2 R}{n^2}$$
where $R$ is the Rydberg constant, $Z$ is the atomic number and $n$ is the principal quantum number, where the greater the value of $n$ (the further away the electron is), the greater the value for $E$ and vice-versa. The negative sign is very significant.
If we want to also compare how kinetic energy varies in relation to the potential energy, we can use the virial theorem
$$\langle{T}\rangle= -\frac{1}{2}\langle V \rangle$$
which states that the average values of the electron's kinetic energy is half that of the potential energy. The negative sign this time implying that the kinetic energy of an electron decreases as distance from the nucleus increases. This is consistent.
A: If you stand at the bottom of well you have a lower energy than at the surface surrounding it. The Coulomb potential of a nucleus effectively is such a well, with graded walls, to the electron.
A: When solving the Schroedinger equation, with the potential term corresponding to the eletrostatic interaction between electron and nucleous, we get an infinity (but discrete) number of wave functions (eigenfunctions).
The eigenvalues for the Hamiltonian are the possible energies associated to them.
The square of the eigenfunctions are the probabilities of finding the electrons at some place, and depends on $r$ (distance to the nucleous) and for some of them also from the orientation (angles $\theta$ and $\phi$). If we make a graph we can see indeed that for wavefunctions associated to more energy (less negative) the probability of finding the electron far from the nucleous increases.
It is similar to what happens in the classical situation, of a test charge closer or far from a spherical charge. The potential energy is bigger (less negative) for greater distances. But the exact dependency from that distance is not the same.

