I have created a custom content type at the root of my Site Collection and set this content type to use a Word document template, which resides in a different document library on the root site. (The content type inherits from Document).
I then created a new document library and set this library to use my custom content type, as described here. The content type is associated with the new document library, and Word opens the template on my local machine when choosing "New Document"->"Content type name" from the Ribbon menu. However, after making changes to the template and clicking Save, I only get an option to save the file locally - Microsoft Word does not upload the file to the document library, which is what I was expecting. Does anyone know what is wrong?
I am using Microsoft Office Plus Professional 2010, if this makes any difference. Additionally, if I choose "File" -> "Save & Send" -> Save to SharePoint and choose the correct library from "Recent Locations", I get the error message "Could not open (url)".
[December edit: Still haven't found the solution to this issue, although I have seen it in more contexts. It currently occurs when working against an SP server accessed through the Internet, although Word <-> SP integration works fine when just editing a document in a document library. On other SharePoint farms, for example a farm in my domain, everything works as expected.
A few other people also have this problem, and AFAIK no solution exists: Link 1, Link 2. Note that the Desktop Experience Solution did not work, although when Desktop Experience was installed, it did become possible to save a document by entering the specific URL in Word. This was not possible before. The problem might also be related to Trust issues, but note that I have added the offending sites to the trusted zone and turned off Protected View in Office, to no avail.]
[Final edit: It looks like we have a solution. I previously attempted the two answers that were suggested separately. After activating both Desktop Experience on the server and installing the Web Folders update on the client, Office and SP interacts the way they're supposed to. Trusted Sites and Protected View in Office might also be part of the solution. Since it's not possible to split the bounty, I'll just toss a coin. Thanks to everyone for the input ;)]