How to clean wood stain drippings from rough spackle wall
You might get the stains off by power washing, but that may knock some of the stucco texture off as well. If you're willing to give it a try, use (rent?*) a power washer with an adjustable power setting and start with the lowest setting available.
So long as it's not causing damage, and since it's probably not getting the stain off, go ahead and wash the entire wall in preparation for repainting. That's probably going to be your best bet in this situation anyway - all the concerns about your clean up methods are very legitimate and are likely going to happen.
Repainting the entire wall has the added advantages of ensuring longer life for the wall by renewing its protective coat of paint, covering all the staining_ so it looks better all around, and after 10 years, the last coat of paint is likely nearing the end of its life anyway.
ProTip™: Before power washing and painting, get some heavy plastic (6 mil or so) and tape it up to the wooden fencing at the top of the wall. Run it up the wall at least 12&quot; if not more. Tape it very securely at the bottom, then leave it a bit loose before thoroughly taping up the top. This way you're not coming back in a few weeks asking how to restain a portion of your fence where you power washed through the surface and/or how to get paint off your freshly stained wooden fence.
Oh, also, use some of that heavy plastic to wrap up the bushes, then some string/rope/tape to hold the plastic in place. You don't want the power washer tearing up your plants, and you probably don't want to be painting them, either. The solution for paint on the plants is much easier though: trim off the painted bits. ;) Also, the plastic will hold in heat and moisture, so don't leave the plants wrapped up for too long or you might kill them.

*Before assuming a power washer is too expensive to purchase, look around and compare purchase prices vs rentals. We ended up purchasing an electric powered one for about the same cost as renting for a day. No, it's not the highest quality piece of equipment ever built, but we got the same day's use out of it and if when we need to use it again, it will now be cheaper than two rentals.