Purchased AE template which has everything set up as "slides"; how to move slide to other AE project

If you duplicate a comp in the Project panel you get a new unique copy of that comp. If you duplicate a nested comp in a timeline you just get another copy of that comp so any changes made to the comp will appear in both copies. 

In slide 10 there are 4 nested comps. If you duplicate Slide 10 in the Comp panel you'll also need to duplicate those 4 nested comps, then open the Slide 10 Copy and replace the Slide 10 Mask, Logo, Elements and background layers with the Slide 10 Copy Mask, logo, elements and background comps you duplicated in the Project Panel.

Complex projects quickly become very difficult to handle manually, but there is an easy solution. It's called True Comp Duplicator and it is one of the best $25 I ever spent. 

What you are trying to do is extremely complex and requires a very good understanding of the structure of your compositions. Even when I work on my own projects things quickly get very complicated. This is the flow chart from one of my recent composites. The average comp in this project was under 7 seconds and look at the mess it is digging through and find the layer with the curves adjustment that needs a little tweaking to fix a glitch in four frames in the final render.

A double click on the layer with Set Matte and Curves will open up the comp and take me to the layer so I can make the edit. There were a total of 6 scenes in the project, 19 main comps, and 137 nested comps in the project. Without the flow chart, it would take a lot of poking around to figure out which layer needed to be edited.

It is possible to pick one of the Scene Comps, Duplicate it and it's nested comps, then move it around and use the duplicate somewhere else, you just have to completely understand how things are linked.

I think what you ask for is not possible, so unless you rebuild the slide 10 in your new project, especially if there is an expression and so, even if you have a project and you want to import the same project inside it, you can't remove the nested compositions from the imported project. especially on the template where there is some controller layers 

Unfortunately, your screenshots don't tell us much about your project. In the future please embed them using the toolbar. There is a little icon that looks like a picture of a mountain range.

Those layers named Slide are compositions that are nested. Everything that is in the Project Panel is an asset. Assets can be sound files, video files, images, solids, C4D files, and folders. I don't think I have forgotten anything. 

If you delete a layer in a timeline, the asset is still there unless that layer is a Shape layer or a Text layer. Text and Shape layers exist only in the timeline. 

If you delete an asset in the Project Panel, it is gone and any layer that once contained that asset will still be in the timeline, but the asset will be replaced with color bars and a missing file warning. 

If you just deleted Slide layers from the timeline you can just drop them back in the timeline as long as they did not have any animated or modified properties. If the layers have any modified properties, which you can find by pressing the U key twice, then you will have to add those modifications back in the timeline.

The way to understand a template is to open up the Flow chart and expand everything. A flowchart will give you the effects that are used and allow you to jump to any comp, or layer or asset source in the footage panel by simply clicking on the thing you want to inspect.

Before we can tell you how to add more "Slides" to your comp, we need to know what is going on with those layers in the timeline. It would also help if we knew what template you were looking at.