@RufusBrewer possible to see photos or links to items mentioned in your post? Thanks.
NPT = National Pipe Thread. Home breweries are almost all be 1/2" NPT thread. It is a series of plumbing pieces that have male for female threads and you screw them together. Make sure you use a generous amount of Tefllon tape. You can purchase a combination of pipe, nipples, T-connections, cross and elbows.
Examples
A T-connection you can ma a 2x1 (splitter) or a 1x2 (Y connection).
A cross connection can make a 3x1 or a 1x3
To make a 4x1 or 1x4 you will have to get creative.
The pipe material of choice in a home brewery is stainless steel. There are also brass fittings. If you want to see what I am talking about, Home Depot and Lowes carries different NPT pipes fittings. I doubt you will find stainless steel at HD or Lowes. But you can get a feel for how NPT parts fit together.
McMaster Carr is an excellent resource to see what is available, and their prices are good enough to give you some budget numbers.
Check out the McMaster page below. It is a good place to start.
https://www.mcmaster.com/standard-pipe-fittings/pipe-size~1-2/
Scroll down the left side to see the Shapes section.
Here is stainlessbrewing's page on what they have to offer.
https://www.stainlessbrewing.com/Threaded-fittings_c_1.html
Other choices are copper plumbing up your system. Instead of threads to hold components together, you solder various T-connections, cross connections, elbows and bends. A trip to Home Depot will give you more insight than I can.
An alternative to copper is High temp PVC. PVC is easy to cut and basically you glue it together.
At some point you are going to decide how to control all the input and outputs. Select some sort of valve design. Manual valves come to mind first. referring to your Step diagram, you will have to come up up with some way to reconfigure your rig to satisfy a given step. Example: All valves closed, except Valve 1, 4, 5, 8 & 12 open.
https://www.dudadiesel.com/search.php?query=SSvalve
They voltage controlled valves are an option. That would be fun to figure out how to control 15 electric valves. Sorry, I am not much help on electric control valves. Here is a page to get you started.
https://www.dudadiesel.com/search.php?query=zerop
I expect that you will end up mounting the various fitting and valves on hunks of plywood, with pipe or tubing connecting to the kettle and pump ports. You do not want a bunch of piping hanging off you kettle port and applying diving board like stress on the fitting.
I would invest in quick disconnects and silicon tubing (not hard plumbing) to connect longer runs, like pump to kettle. I want ti see water and wort moving.
Where to get stuff? You are going to be buying up a bunch of parts and materials. Shop around, buy in bulk, get quantity discounts. Try eBay and Alibaba to see what you can get at reduces prices.
Hope this helps.