Add lightning component to a navigation menu of community site

You need to have the component implement the interface as follows:

<aura:component implements="force:appHostable" />
//...your code here
</aura:component>

This is what tells the platform that you wish to use it in the S1 mobile application.

You can find more about this in the Lightning Components Developer guide in the section on Adding Lightning Components to Salesforce1.

The Invoice object probably isn't on the list because it's part of the managed package which hasn't been released for use in a Community. Only standard objects that are available for the community type and custom objects you've created will be visible along with managed package objects that have been released for use in Communities. Check the managed package to see if it's available for use in a Community. You may need a separate license from the vendor to use it in your Community.