The flowchart is your friend here. You'll have to load up all of the templates, figure out how they work by looking at the flow chart, then start a new comp and copy and paste the elements you want to use in your new comp in the timeline. Once you have everything in place you can use the File>Dependencies>Reduce Project
If you don't know how to use the flow chart check the help files by typing flow chart in the search help field. When I get or create complex projects I'm referring to the flow chart all the time. It's a very useful tool that can take you directly to any nested comp or nested layer in a project.
It's tough to know what you'll be able to glean from a flowchart without an excellent memory and a lot of painstaking work. Templates are notoriously complex -- if they weren't, there would be no reason to buy them. And there are NO standards for constructing an AE project. So you have your work cut out for you.
The mini flowchart is handy for a quick look but you are better off looking at the flowchart panel and having it next to the composition panel. To figure out what parts of the template you want to keep you drill down through your timeline until you find the part you want to keep. Let's take this simple example:
From the main comp I want the top line of text and I want the shape layer with the text overlay that says "want the shape." Imagine that this comp was as complex as yours. To find the parts that I want to use in my new comp I can start clicking in the flow chart. A single click on any of the flow chart icon's selects the comp in the project panel so you know what nested comp is being used or if the asset you want is a layer in a comp the layer is selected if the nested comp is active. You can make any layer or nested comp active by double clicking the asset in the flowchart. This helps you quickly figure out what part of the template you want to use in the new composition.
It also works the other way around. Say you have a very complex project, like your example and you want to use layer 20. If you click on layer 20 it will highlight it in the flowchart. Even if you open up a nested comp, as long as the flowchart is visible, clicking on any layer in any flow chart will highlight that layer in the flowchart if the chart is expanded.
When nested comps are selected in the project panel you can just drag them from the project panel into your new comp. I usually do all of this in the Project panel and then arrange everything later. If you want an individual layer that is part of one of the nested comps like the shape layer withy the text in my example you drill down to that asset, select it in the timeline, copy it and then open your new master comp and paste it in.
Once you are done you just run the Reduce Project utility from the File Menu and you have everything you need. This is a lot faster than using the "locate in project" feature from a pop-up menu. The mini flow chart doesn't help that much because you can't do any clicking and locating in the Project panel. About the only thing it is good for is quickly opening or navigating to a nested comp. Still handy but not nearly as useful for picking and choosing assets you want to use in a new project.
Were you ever able to pull the elements you needed from this template? Did any of the advice from our experts help you out? Please let us know what you did in solving this issue, as it would definitely help another lost soul with the same issue.