OP, the reality of the situation based on your parameters is:
Move.
Just kidding. Seriously, as the owner of a mature red oak and the former owner of a crepe myrtle (full sized), neither of those species are ideal for that spot based on your requirements and the photo.
Red oaks are wonderful trees and we love ours. It does provide great shade but allows the grass to grow underneath. However, there are some very good sized roots, a lot of leaves and these things called acorns that would create several problems with your location. Some would be a nuisance, but the roots and the retaining wall would be $$ even if you put in a root barrier. If you put in a root barrier to save the retaining wall, then those roots might spread to your foundation, also $$. Which brings me to the crepe myrtle.
A full sized crepe myrtle has larger roots than you might think. When we removed ours it was in a bed in front of the house and per the norm, the builder had placed it about two feet from the foundation. That is not far enough! Those roots were going under the foundation $$, etc and I was amazed at just how large they were and what they were able to do. The leaf litter and blooms from a crepe myrtle are not quite as abundant as an oak, but it will get in your gutters if not sealed.
I have little knowledge of the trees such as the cypress mentioned, but perhaps you can locate someone who has a mature one and ask a few questions about it. Look for examples around your city to see if it may work. A perennial vine may work, but again, for that you would need a structure or support that is stable enough to support the vine but not wreck the wall. That wall looks like an important feature for both the houses with respect to drainage, etc and that would be where the headaches would come in if it were to be compromised.
If energy efficiency, privacy and shade are what you are looking for, perhaps see if someone from the utility department can visit and do an audit for you because they may have some suggestions as well.
Just my two cents.
Best of luck to you!
