I found the solution!
The problem seems to be with the new version of Origin. So you need to follow the steps set out by DMEX in this thread:
http://forum.ea.com/...15/8633365.page
First, manually uninstall Origin. As in go to the folders and delete all files in Program Files AND ProgramData folders relating to Origin. Make sure to also delete all registry entries. To see how to do all this, see:
https://help.ea.com/...stalling-origin
Then install Origin from the disc but DON'T update when it asks you to update to 8.6. Rather, go to the DMEX thread at the top and download the older version of Origin files, the 8.5.2.23 ones. These you must then extract and copy over to your Origin install folder, overwriting everything there.
Next, go to you Origin shortcut and do as DMEX shows on the image, in the target line add /noupdate. NOTE: he left out the quotation marks, so your should look like this if you used the default folder:
"C:\\Program Files\\Origin\\Origin.exe" /noupdate
Now when you run Origin, it'll be the older version without crappy errors and will not try to update. You can you install Mass Effect and it'll complete. Give it a while though, mine still sat on 99% completing for a little while and I nearly lost my mind, but then it finished and updated the game and all was well.
Remember, this is a temporary solution to allow you to install the game. So once you have done that, after a few days just remove the /noupdate command and allow Origin to update to the latest version. Let me know if this works for you. Good luck!