Why should a high energy electron have to remain in a deep potential well?
As long as the electron energy is lower than the well, the electron will remain bound. If you are thinking of the infinite potential well, "high energy" is still less than infinity.
What if the electron has an energy above the highest bound-state energy but below the top of the well?
It cannot. There is no such eigenstate of the Hamiltonian. It might have such an expectation value for the energy. Then it needs to be in a superposition of bound and scattering states.
Why an electron having energy above bound state but below top of the well will have no eigenstate of the Hamiltonian? If it has no such energy eigenstate how can it have energy expectation value?
In order to have a definite energy, a state needs to be an eigenstate of the Hamiltonian. There are states which are superpositions of different eigenstates. These states have some probabilities of being in the different eigenstates, meaning that the expectation of the energy does not have to be an energy of an eigenstate.
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