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The little arrows mean things are clipped. as in the only affect the item immediate below the layer stack.
If We go back to the first image you talked about, if you double click the layer it will open a new photoshop file and that photoshop file will have all of the layers that make that one graphic composition.
The photoshop file will now look like the second one. You can take that file, save as, and save it somewhere. You can start doing this as you find files like the first one so you can now only have the one photoshop file with all of your working layers ready for editing. that is how i work. I personally think you want to work that way since is less clutter and you always have the working file ready for editing.
You make a "smart layer" by placing the photoshop file in a layer of another photoshop file. Which is what the designer was doing to make you the screenshots you wanted? i guess?
I don't know of an automated way for photoshop to go to every single photoshop file in folder and find and perform the tasks you asked for all automated.
Here is a link to how to work with smart objects if you want to understand better how they work;
I actually want the opposite. If I click into the first image with the dropper it looks like the second image, I get it that I am going into the smart image. I want the second file to be like the first where I have to go inside the smart image to make changes. Since we rarely make changes it's easier for what I need it for to have it like the first image, all compressed into the smart image. I just need to know how to change it to make it that way.