First and foremost check the H: drive for .tmp files (more importantly ~*.*)
I've run into this quite a bit where ~* files on the local and server are causing Word to take an in ordinant amount of time to access the info...

If that doesn't resolve it...
This document is attached to a template on the K: drive - Per your instructions I want to ensure that we are not making a local copy of the template before creating the new document, but making a new document, based off of a template on the K: drive - then saving the file locally (Is this correct?)

I'd be tempted to see what would happen if the template (any template from K: drive) was saved locally and the originating doc came from that template instead of the server-based one...
If, after copying the file up to the H: drive (the one based on the local template) - it opens more quickly - then we know we have something wrong between the server templates and H: drive stored documents and can troubleshoot that way...
if its still slow - we've completely ruled out the templates on the K: drive...

Let me know what you find,