How to create links from term to node
The module Special menu items lets you create menu items that aren't links. 

Using Taxonomy doesn't seem like it would do anything to solve this, unless you want a different list of links to show depending on what node it's being displayed on. 

So basically you just create a new Menu and add menu items you want in the list. A Block showing the list is automatically created for each new Menu, so you just have to go to Block configuration and place it in a region of your theme to make it appear. On Block configuration you'll also be able to restrict which pages to display the links on, maybe just the one node.   

The thing you asked about using NID is concerning Path aliases. You can use the module Global redirect to make the system always output the alias instead of the NID link. With Pathauto you can configure patterns to automatically create path aliases when creating new content.