Is it possible to create a calendar so that it appears within a specific Folder in the Documents Library? I do not want it on the main page under "Lists."
You can spoof it through the UI (Site Actions -> Site Settings -> Quick Launch (or Navigation depending on your enabled site features), you can manage the quick launch/side bar so it shows up under documents heading.
No, you cannot have a list within a list (document libraries are also lists).
Depending on your exact needs, this might be a spot for a Document Workspace.
"A Document Workspace site helps you to coordinate the development of one or more related documents with other people. The site provides tools to share and update files and to keep people informed about the status of those files."