If your content wont work offline its because its not licenced to that particular 360 console. So what you did not do was download it on your 360 with your hard-drive attached. You must have downloaded it on your "friends" console with your hdd attached to it, which is a rookie error.
All 360 consoles have unique hardware signatures for both the console AND the hard drive. If they do not match for any reason the game title/DLC will not work offline for any reason. You will need to do a "licence transfer" and register it to your current console.
http://support.xbox.com/en-GB/xbox-live/marketplace-and-purchasing/download-content
And also you do NOT require an xbox live gold account to download content. So that "free month trial" wasnt required at all.