How do I alter the path of an uploaded image/file such that it includes the path of the node I attach it to via an image field?
Seems you are looking for a File (Field) Paths module:

The File (Field) Paths module extends the default functionality of Drupal's core Upload module, the FileField module and many other File Upload modules by adding the ability to use node tokens in destination paths and filenames.
In simple terms, File (Field) Paths allows you to automatically sort and rename your uploaded files using token based replacement patterns to maintain a nice clean filesystem.

Be aware that it's not yet stable. Test intensively before using in production environment. On the other hand, I don't know anything better and I know for a fact it may work all right.