Can I copy my friend's game to my Origin? [we both own it]

Yup, you can copy across the game files without issue. Ownership of the game is based on your accounts, so there's no problem with sharing the actual game files.

This backup/restore guide should help with the specifics, but it's a pretty easy process: http://answers.ea.com/t5/Origin/HOW-TO-Backup-restore-Origin-games/td-p/1926893