Hi notist,
If you are able to write down the perturbed Hamiltonian, you should be able to run through these computations quite easily :).
The idea is that to first order perturbation, the energy shifts are essentially the same as the expectation value of the perturbing Hamiltonian. It seems like that's what you are trying to evaluate in the first problem. In the textbooks you have, it gives the simple formula for this.
You have the second order perturbation in that image you added. I think you should see its pretty straightforward, once you realize what that matrix element is. The H' you see there is your perturbing Hamiltonian (in your case this is just E(x)*x, and the psis are your regular old harmonic oscillator wavefunctions. Notice that the superscripts on all the perturbation theory things are indicating which order you will use. For energies, you never need to worry about the higher order wavefunctions.
For the last problem, you are going to need to get the wavefunction correction for higher orders. You can look this up in a book, its the same sort of idea. Because you note in (b) that the second order and higher will be as good as the first, you only need the first order. This is just an exercise in finding an expectation value, with the lowest order wavefunction, then again with the higher order wavefunction, and comparing them.
Hopefully when you don't have to worry about homework, you can really sit down and figure out how perturbation theory works. Its a very good tool!
