If you are lucky, the missing strainer may not be end up having caused too much of a problem. It is there as a backup to your filter, which should be catching most of what makes it through your pump basket. (Did you say what kind of filter you have?)
I don't think I saw an answer to your question about the manual valve after the filter (8th picture in the post before this.) That valve is controlling the return to either pool (left side) or spa (right side). In the picture, the diverter is in the middle, so that it is roughly 50/50 pool/spa, depending on downstream resistance. For your in-floor to work best, you probably want to close off return to the spa (OFF facing the right, handle hanging over the "Pool" writing.)
The return "Pool and xxxEP" (did it used to say Sweep?) pipe should go into the ground and come up next to the Caretaker. Then you have another manual valve that can give 100% to the Caretaker. That combination is most likely the best configuration for the Caretaker operation.
Side note, what kind of pump do you have - single speed, dual speed, variable speed? Pump settings may be important in dialing in the right settings for your Caretaker.
But also warning you now that there are many opinions from in-floor system owners. Some people here have abandoned it for cleaning and gone to a robot, both from a cost and effectiveness point of view.
The last step downstream to get your Caretaker working is to make sure the pop-ups are functional. They need enough pressure beneath them, but the spring in them can also get cranky, and if any debris made it in there, that can clog the water pressure underneath.
I would do this step last, but if you are purchasing parts from an online site, consider also purchasing a pop-up removal tool. This
plastic removal tool is basic and inexpensive, but if your pop-ups are stuck, it may not work. If it doesn't, then you can purchase the more expensive and sturdy
metal removal tool. Either way, please note: if your Caretaker pop-ups are 1992 or older, they are reverse threaded, so they unscrew clockwise.
One last comment: it is surprising to me that you have all these valves without any of them being automated!