That first area sounds like a particularly good area for a swale by creating a dip between the wall and the bermed area where the lawn is now.
Even a permeable walking surface such as a gravel or woodchip terrace can function as collection area for some runoff.
For example that area of rough stepping stones, which ould be made more permaeble wy setting them in a matrix of something like decomposed granite.
Interesting - there used to be a ficus tree there so it's kind of hard to excavate. I'm not sure how we would do that without some intense stump grinding. We'd also have to pull out the irrigation in that area too.
The inside 'courtyard' area, including the flagstones, and that brick-framed rose garden, I would like to flatten out and turn into a more usable space. It would be nice to have a lounging area out there too. Something with this look but also with the pavers wider apart and some areas where native plants are planted in the sand/areas between the pavers:
The problem is that my wife doesn't like the flag stone look very much, so we would have to find something we could both agree on. I do want to keep the orange tree as well as the other two trees in the furthest part of the courtyard (where the catch basin is) - one is a peach tree and I'm not sure what the other is but I think it's another type of stone fruit (just checked - it's an apricot tree!). We also have a crab apple tree, loquat tree (tried one from another plant we found walking through our neighborhood and they are delicious! our tree actually has some sprouted but they're super high up - I need a coffee can/tin on a long stick to collect), and some other type of citrus tree along the side near the backyard. None of those are doing that well, so if they can't be revived, we may just removed them. There's a large [fruitless] avocado (I think) tree as well as a fig tree in the side alley too, along with a couple magnolia trees and holly trees. I like the magnolias but the others may need to go. I'm not sure what we would do around all those trees and if it's better just to pour in some small and medium sized river rocks and mulch the base of the trees we want to keep. Our yard is actually a decent size if you consider the back, side and front courtyard as one unit. We could theoretically even stretch the section of courtyard in the front where the rose garden is out more forward and enlarge the entire front courtyard and have a smaller front lawn (visible to the street). But that would probably cost quite a bit more money to do.