Try booting into Recovery Partition (Hold Command + R while Booting). then if you have all of your files on the external drive you can restore (reinstall the OS) and then use Migration Assistant to transfer files.1) I believe I'm running either Mountain Lion OS 10.6 or 10.8 (one of them for sure)
2) I don't have any CD Disk (but I do have a dmg installed on my external hard drive) I was wondering if there was any way to use target disk mode to install it or trouble shoot it.
3) I do have my files backed up on the same external drive.
4) What actually got me in the mess in the first place was --- I noticed I had two user accounts. (a
mac repair shop who set this up gave me a new hard drive with my old files on there & set up a new account) I didn't think my computer was using that particular account as the main account and thought that if I put it into the trash, I could simply take if back out. However, it turned out that it was actually my main account. Then it was asking me for my pass-code, but is in the trash.