well looks like you might not of setup your word conversion on central admin or your trying to convert an unsupported file:
You should see somthing like this on central admin to configure your word service:

full tutorial can be found here:
http://www.c-sharpcorner.com/UploadFile/anavijai/configure-word-automation-service-in-sharepoint-2013/
I would think that contribute or designer is about right for permission as the actual conversion permission is based on the webapplications app pool for word conversion not the user account, so its more about being able to change design ect on the page as your converting the doc to show as html page within the content area.
if the above fails there is somthing else you could try, i have found a blog that notes that it could be set to run under sandbox mode:
after SP1 was released, the Word Automation Service had been set to
run in Sandboxed mode. What does that mean, and why would Microsoft
change it, I don't know, but I'd love to find out. So he proposed
disabling this by running the following PowerShell commands:
$sp = Get-SPServiceApplication | where {$_.TypeName.Equals("Word Automation Services")}
$sp.DisableSandbox = $true
$sp.Update()
http://www.chrisbuchanan.ca/Blog/Lists/Posts/Post.aspx?ID=23
EDIT
also like to note:
•Turning off the Document Conversions Load Balancer Service stops the
document conversion process and turns off the Document Conversions
Launcher Service. To resume the document conversion process, you must
restart the Document Conversions Load Balancer Service and the
Document Conversions Launcher Service, in that sequence.
•The Document Conversions Load Balancer Service and the Document
Conversions Launcher Service are often run on application servers, not
front-end Web servers. Wherever they are run, the services must be
configured to communicate with the front-end Web servers, as described
in this article.
1.Verify that you have the following administrative credentials: you must be a member of the Farm Administrators group on the computer
running the SharePoint Central Administration Web site.
2.In Central Administration, on the Home page, in the System Settings section, click Manage services on server.
3.On the Services on Server page, in the Action column, click Start for either the Document Conversions Launcher Service or the Document
Conversions Load Balancer Service.
You must configure and start the Document Conversions Load Balancer service before you can configure the Document Conversions Launcher service.
4.If you click Document Conversions Load Balancer Service, the Load Balancer service is started.
5.If you click Document Conversions Launcher Service, the Launcher Service Settings page is displayed.
6.In the Select Server section, verify that the server you want is selected. If it is not selected, on the Server menu, click Change
Server. Next, on the Select Server page, click the Name of the server
for which you want to configure document conversion settings.
7.In the Load Balancer section, select the load balancer server to use with the launcher. This is the name of the server on which you
configured the Document Conversions Load Balancer Service (see
previous procedure.)
8.In the Port Number section, type the port number that you want the launcher to use for communication. Enter a port number that is not
used by other services on the server, but that is open in the firewall
on the server.
If the Document Conversions Load Balancer Service and the Document Conversions Launcher Service are on the same server, the port numbers used for each service must be different.
9.Click OK. The Launcher service is started.
http://technet.microsoft.com/en-us/library/cc262093.aspx