Since we know nothing about the project's specific requirements and its structure, we can't tell you whether this is an issue or by design. Judging from the mess in your screenshots, I'd say you simply need to learn how AE works instead of hoping to play it quick & dirty based on a template that you do not understand.
There's no way to tell.
There are bazillions of templates out there, and the good ones tend to get really complex -- lots of nested comps, expressions, etc. and you have to follow the instructions that come with the template to make modifications. By the way -- templates are typically designed so you can make limitied modifications within the template designer's parameters, and it's easy to do... within varying degrees. You just can't download AE with no prior knowledge of it, download a template and expect success at the first go. It just doesn't work that way.
If you're making simple modifications with the template design modification parameters, your screen shot doesn't show us enough, and you're not telling us enough.
If you're attempting to make modifications that go beyond the template design modification parameters, we have no clue other than to tell you, "learn more about After Effects".
Pretty much what Dave said. It's impossible to tell from your grabs, but at a guess I'd say there's expression linking going on with the layers, so by adding more layers, which is not what the template designer had in mind that you would be doing, those expressions are breaking and causing the camera to freak out...
A better way of putting it would be, you're going to have to learn AE at a more in-depth level if you want to continue with the changes you're hoping for I'm afraid!