OBSE is a simple download that you put in your oblivion folder, and from then on you simply use a different .exe to start the game. It allows you to use mods that need scripts that Oblivion doesn't. (Thus "
OBlivion
Script
Extender")
CM is a type of companion mod interface thingy included, and required for HOD. It is what causes those rings to be in your inventory. If you do not have it, I believe one of the outdated versions might not have needed it, but then that confuses the issues of why you have rings you shouldn't have.
To find the formids for the item you can simply:
start up the Construction Set,
click the folder icon in the top left
check the box for HOD's esp
and then press the button that says "ok" or "load this as my active file" or whatever it says.
From there at the start you should see a window that lists races, items and such. Simply browse through them until you find keys, or quest items or whatever.
Now just hover over the item until the ID appears, and write it down for each of them.
It'd really be much easier to just download the most recent non-beta http://www.google.com/search?btnG=1&q=OBSE&l=1 and then download that mod I linked to earlier.
EDIT:
Won't removing CM partners get rid of the rings? Far as I remember, they're not part of HOD but included with it since the mod requires CM.
In my game CM is a master for HOD, thus removing it would cause crashing.