Let's go from left to right, blue-handled valves 1, 2, 3. I agree, #1 should always be wide open. I can't explain why it wasn't. Frankly, that valve shouldn't even exist, as it will be one of the first to fail (ball valves are awful in that regard, and can't be repaired), and there's nothing it can do that the black-handled three-way valve can't accomplish. Same for #2. Redundant, useless and failure-prone.
The only reason for those valves to be there is if your equipment pad is lower than your pool's surface. So maybe that's the case, as I see them on all your pipes. It might have something to do with your negative edge, as in essence you have two pool levels, three if you count the spa. I digress.
BTW, the "flapper valve" is called a check valve, and it's purpose is to allow water to flow in only one direction. There is an arrow on its clear lid that indicates the direction, as does the black and white arrow label on that same pipe. The lid is clear just for this purpose, to determine when water is moving through the check. So...
You say in pool mode, that the flapper indicates water is flowing, and that you have water spilling over from the spa to the pool. In spa mode, no spillover and water circulates only through the spa. The latter is normal, the former is not. Your pool should have three modes:
- pool mode (no spillover),
- spa mode (no spillover), and
- spillover mode.
It sounds like you have no true pool-only mode, which is why I asked about it. Spillover mode should be a function of your automation, selectable when you want it (and/or schedulable), and not on full-time whenever in pool mode. The reason you're getting spillover when in pool-only mode, is because valve #3 is wide open.
There's nothing particularly harmful about constant spillover. Sometimes that can contribute to rising pH. And sometimes people don't want to listen to their spillover, and then just want it nice and quiet. Which is why it should be selectable by automation, not necessarily on all the time.
So if you'd like to have those three different modes, you need to shut valve #3 and make sure your automation is setup to give you spillover mode.
Now, why would that valve be there? Well, it's to allow a trickle of water to flow from the spa to the pool, even when in pool-only mode. That's to allow some water to always be moving between the two bodies, to keep the chlorinated pool water circulating through the spa, to keep it chlorinated. This is useful when you don't use full spillover rarely or ever, or don't use the spa much or ever. You either have to have your automation schedule some spillover each day, like 1/2 hour or so, or you use this crossover pipe to run a small amount of flow between the two bodies, all day long.
Either way is fine, and it's nice your setup gives you all the options. I called this to your attention so that you could choose the options that work best for you. And to point out that the crossover valve is not usually wide open like that, keeping the spa spilling over fully when in pool mode, whether you want it to or not.