Each site or subsite on a server (domain name) shares some files and has some unique files.
Each server has only one template-assets folder which all sites on the server reference. These files should never be altered except by upgrades to the template package. All sites on a server will reference at least some of these files.
Each site on a server should have its template-assets-custom folder located at the root of the site. eg: /subsite/template-assets-custom/.
Existing sites probably already have this folder. In this case, just the 07 subfolder needs to be added. eg: /subsite/template-assets-custom/07/
Templates (includes, images, styles) customised for the subsite can be changed to reference assets in the new custom location.
The Dreamweaver templates are not required on the server. They are only needed to format the local files on your computer.
On the other hand, uploading templates to the server is a good way of sharing them among maintainers. All templates for all sites go in the Templates folder at the root of the server. If you are uploading templates to the server, you must rename your templates so that they do not replace other templates already uploaded.
Instead of leaving your template as subsite_content_07.dwt you should give it a unique name that allows other users to readily identify it. eg: mysite_content_07.dwt
Maintainers of subsites sometimes do not have permission to put files into the Templates folder. If this is imperative for sharing, your site administrator may be able to upload them for you.