Chris, can you try the no configuration, configuration? Instead of manually adding the server information, just download and install DTI. Now log into WC and click on an Office icon in the browser. With the Download Mechanism set to DTI, this will launch DTI and auto-configure it for the server you are attempting to download the file from.
Yes, we do leave the regeistry settings behind during un-install. It is basically the connection and client configuration information. You can manually remove this information from the HKEY_CURRENT_USER\Software\PTC\WindowsIntegration key in the registry.
PTC has a backlog story to update the installer to include removing the user settings for uninstall, but it has not been prioritized it to the top of the stack yet. Hopefully it will be queued for the release following 10.2.