Add-on system installs packages that have EPiServerPublicModulePackage or EPiServerModulePackage tags in nuspec file. That's why it probably won't install anything from the NuGet feed.

Adding the official Nuget feed as another repository is not an option if you are going to distribute your add-on in EPiServer Store, since most of the sites will not have NuGet feed repository configured.

Are you going to place your add-on to Third party section of official Add-on Store hosted by EPiServer?

There is no way to have a dependency on the NuGet package that is not an Add-on, but you can have a dependecy on a library that is a part of the NuGet package. The Add-On system treats every library in the site \bin folder as a virtual package. If you want to have a dependency on, lets say WebActivator, you can just add this to you add-on dependencies:

<dependency id="WebActivator" version="1.5.2" />

The Add-on system will check if there is an Add-on with ID WebActivator (apparently there is not) or a library in the site \bin folder with name WebActivator and verison 1.5.2, before allowing to install your addon on the site. So your Add-on will be available only on those sites that have a WebActivtor already installed.