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You need a landscape designer who can help you deal with all this and perhaps choose a mix of hardscaping and landscaping suited to your conditions of alternating heat and wet conditions.

Anything to hide the fencing. For example

Rosemary hedge (rosemary is indestructible in your/our area). There is a prostrate rosemary that will grow down to partially cover the wall.

Texas purple sage (not a real sage)

Lantana

Various hollies

That is very weird that the grass did seem to grow in the shade of the house but not out in the sun. It looks like it never got water, but I know for a fact that can't be true. I got 20 inches in my gauge for September.

some ideas:

What is the depth, front to back, of the space at the top of the wall, between the wall and the fence?

In your area, spring may not be the best time to plant since you will be going into the hottest, driest season. Fall into winter may be a better time for many plants so that they can develop more roots before heat arrives.

Is your goal to create a completely enclosed sanctuary, or are you happy with keeping open views of surrounding houses as in DigDoug's design above? How important is preserving sun, for both the yard and house interior? I think your choice of groundcover will follow from this basic decision.