My neighbor gave me an amaryllis in a pot with no drainage hole, and it has bloomed for 3 years in a row. I don't get it. I just water when it gets dry, but I don't let it out in rain to get soaked. I keep it under a covered porch or carport in summer. And bring it in for winter. I had several plants survive for about a year after my mom died, sent in a nice hole-less pot but I repotted them after a couple years---pothos, coffee, dracaena, prayer plant. I wish I could use all those pots without getting out a drill, but they are too shallow. But I do tend to root plants in water and they seem happy for a year or so at least. So maybe those--anything but succulents--won't mind these kinds of pots? I have all kinds of plants rooting and growing in water.

If you get a cactus and succulent soil from one of those places, you'll need to locate large perlite. The only perlute sold at those stores is small and difficult to deal with. Alternatively, you can use large chicken (granite not oyster) grit bought at a feed store but this will make the pot heavier than if you can locate large perlite. I get mine at the organic and hydroponic store. You could also substitute leca or seramis or another hydroponic media. 60-75% of these additives and the rest cacti/succulent mix.

Talk to your town's planner or code enforcement officer and find out what the rules are. Get a copy of the neighbor's building permit (It's a public record), probably available on line, as is the building code. You may need to talk to an attorney. But between the rules and their permit, you may find the answers. You want to think clearly about your proposed remedy if they are violating the rules, don't settle for something the propose. (Among other things, make sure the pony wall is on their side of the lot line.) If the building code is incredibly long, don't be intimidated. In the city nearest me, it's nearly 1000 pages, but most of it is highly repetitive. They repeat the same definitions in every zone and you only have to read about your zone.

Greta idea and very doable. I do something similar but just the main pot dug into the ground. The drainage properties can become worse, remain same or actually even better depending on your yard soil characteristics. You may find the top few inches drain well but as you go down the soil is harder and more compacted. You want to stay above the very hard layer. To find out dig a few holes to various depths and fill the hole with water and measure how long it takes to drain completely. Also measure after a hard rain. The deeper the hole the longer it may take to drain. In our case, the land is elevated from our surrounding and 12-18 inches deep holes drains pretty quickly even after hard rain. When I say quickly I mean less than 30 min in our yard. Yard drainage specialists use similar methods to get an idea. They may dig a deeper hole say 2ft. And if it drains in 4 hrs or so then it is considered good but I think it is a bit too long for a dug in planter. I would not go that far. I have my pots dug in 4-12 inches. A large pot may still stick out but gives a lot of stability. That is one advantage. Second, not so intuitive, is that it improves the drainage property of the planter. This works if the soil below is in intimate contact with soil in the planter. The earth acts as a giant wick and sucks out excess perched water from the planter, if any. The downside is that you may find out thick roots have made their way into the soil. And if left like that then it can block all the drainage holes. I drill many holes in the pot, sometimes even on the sides. The roots will tend to prefer the side holes or holes near the bottom edge to escape into the soil. If you are doing pot in a pot consider having the outer pot bottomless. Or add more holes and fill the bottom inch with native dirt. If you do not add the dirt, it will still work, but drainage property of the pot would remain unchanged. It will behave the same way as if it was set on the ground or patio. But if the roots make it to the ground you may find the pot-in-pot to be more trouble. If all goes well, you will find that your plants will be happier since the roots will be cooler. The plant will be stable. And as you said it makes it easier to "redesign" the garden. I use one pot and adjust the hole size. I have my potted bulbs like this. And when they are done, I will put in my tropicals. I even have pots partially buried in limestone gravel beds or mulch beds. By the end of the season I end up with more roots in the gravel or mulch than in the pot. And big vigorous plants.