Ok, here's the rest of my reply to your questions:

Do I have to calculate 2 different values for my E(Ψ) because if you look at the assignment, there are different potential values for V = 0 or 1

At this point it's just a math problem. You are essentially evaluating

[tex]\int_0^L \psi^*\hat{H}\psi dx[/tex]

As you've noted, the Hamiltonian includes a goofy discontinuous function. Nevertheless, the problem is basically set up the same as the case where the potential term V(x) was continuous over the limits x = 0 to x = L. The only difference is that here you have to integrate over a discontinuity. Integrating over a discontinuous function just requires breaking the integral up into smaller integrals, one each for the three continuous regions, each with integration limits at the points of discontinuity.

E.g. consider a discontinuous step function with point of discontinuity at x = b:

[tex] \begin{equation*}

f(x) = \left\{

\begin{array}{ll}

g(x) & \quad a < x < b \\

h(x) & \quad b < x < c

\end{array}

\right.

\end{equation*}

[/tex]

Then the integral over the range x = a to x = c is

[tex]\int_a^c f(x) dx = \int_a^b g(x) dx + \int_b^c h(x) dx [/tex]

This works for continuous functions as well.

Anyway, hope that helps. Feel free to hit me again if you are still having trouble.