Your pictures are too close up and don't show how one piece of the back yard fits with another. The main question I had after reviewing your post is, where is all the water supposed to go? In that hole? Is that going to be a french drain? I don't have enough information to summarize the project, but my suspicion is that this project is not well thought out and you're on a path to overkill... making the project bigger (and more expensive) than it could otherwise be. With a drainage issue, what must be understood is the PATH that the water will take which clearly shows that it can exit your property without obstruction. I'm guessing that instead of solving the problem, you're making a "bowl" further away from the house to put the water in. For most rain events it will probably work (but not necessarily without problems.) But when that torrential, freak downpour occurs, the water will fill up the bowl and come back to you. Once the path of the water flow is identified and a strategy is created to direct the water, it may turn out that you can do it without a wall, or with a very low wall. Please put some photos (not close up) on the thread that clearly illustrate the path of water flow until it exits your property. Since it looks like water is entering at the back of the lot, I'm guessing it's leaving at the front. That's where photos should start. Take pictures that illustrate the grade.

No need to put a drainage pipe behind the wall. Make sure you put in a good solid base and use gravel behind each course to stabilize the rocks and reduce washout of soil behind the wall.

Your drain pipe doesn't go below the wall, it goes behind it. This is why you're able to use flexible 4" ABS plastic. Think about the movement of the water - that pipe behind the wall exists as a pressure relief valve, so that if water builds up behind the wall it can be diverted out the face of the wall before the wall is forced over. If you put the pipe below the wall, where will that water go? If it's going in or around the footer, your wall will fail. Look up Anchor retaining walls online and look at their installation details. Even though theirs is a manmade product, the principles of the drainage areas are the same. An 18" garden wall is no big deal. You're talking about a four or even five foot wall? That's nothing to screw around with, especially since the goal is serious water diversion from a basement that's already had water intrusion issues. I can't recommend strongly enough the involvement of a professional.

You need a book; no one can give you advice on this without seeing, and even if they did it would be irresponsible to do so without knowing how you work. So grab the stone, as much as you can get - you can probably get rid of excess if you end up with it - then take your time reading and planning so you do the job right. Remember that stone can also be bought, probably the same kind or similar enough to blend in. So don't panic if you're a few stones short.