This is more of a Windows 7 question than LibreOffice, but my searches on the internet haven’t turned up anything for my situation. Maybe someone here can help.
I have LibreOffice 3.6 installed on my Windows 7 computer. I am not using MS Office at all. However I still exclusively use .doc files for Writer because everyone else I correspond with uses Office.
I want to add the option to create a “new” .doc file to the Windows context menu when right clicking on the desktop or folder empty space.
I think this is a wrong approach. The best practice would be to require your correspondents to use an ISO format and not a proprietary blob. 
You can also choose to use .doc as your default file format. Go to Tools > Options > Load/Save > General. Look at the section “Default file format and ODF settings”, then make your choice.
Until they do that… where can I obtain the plugin so that Office 2003 can open/save .ODF documents? I’ve found the 2007 version here, but no luck yet with 2003.
The second part of my answer fits your first question. This is another problem, and such a plugin doesn’t exist.
I’ve used Sun ODF Plugin for Microsoft Office 3.2 with MS Word 2000 — “System Requirements:
Microsoft Office 2000, Office XP, Office 2003, Office 2007 (Service Pack 1 or higher) or the equivalent stand-alone version of Microsoft Office Word, Excel or PowerPoint.”
See also: Sun ODF Plugin for Microsoft Office 3.2 · readme.txt → http://dl.dropbox.com/u/19264180/LibO/Sun_ODF_Plugin_readme.txt