It is possible to add any files (not only .dll) and even directory structures to a NuGet package, so in theory you could store all files belonging to a project in a package and publish that. However, it is only possible to install NuGet packages into an existing project, meaning that you won't be able to add the packaged files as new project to any solution. In addition, the contained files will be put to the solution's packages
folder, not to a solution or project folder.
As an alternative, consider creating a Template project instead of a NuGet package, as explained in How to: Create Project Templates and my blog post Creating template projects. This will result in a .vsix file that any developer can install as Visual Studio Add-In, the templated project will then appear in Visual Studio's File -> New -> Project... dialog as new project type and can be added to any solution.
Similar to your NuGet gallery, it is also possible to create a private Visual Studio Gallery to share such Add-Ins within, for example, a company, as explained in this MSDN article.