Restored Time Machine to external - now what?

I had a hard drive failure on my iMac during a power outage. All the data seems to be there but it does not boot the OS. I repaired using DiskWarrior and it verifies fine using Disk Utility but it still does not boot. (It actually did reboot after the outage, but was running strangely so I tried to restart and that is when it failed). I had an old system installed on an external so I booted from that and made sure I copied everything from my computer over just in case Time Machine missed something. I didn't have any trouble accessing any files.

I have a Time Machine backup from before the power outage so I installed that onto a different clean external drive. It is working fine outside of a previous issue I had, which I have now solved. I have been working from this external for two days now.

My question now is, how do I backup this external onto the same Time Machine backup that my iMac hard drive was backing up to (and that was restored onto this external)? I am hesitant to turn Time Machine on because I do not want to mess with my old backups, or have two full backups. I just want to backup everything htat has been added or changed since installing the new version on Monday. What I want to do is create a better version of my system on Time Machine that I can then restore back onto my internal hard drive.

I am also wondering if there is anything I can copy back to my internal to make it actually boot, since all the files are still ok?

Should I take this opportunity to install Yosemite on my iMac? Can I install a clean version then restore my files from Time Machine? I'm not sure how upgrading that works. There is a lot of old stuff on my computer that I probably don't need, but I also do not want to start completely over because of preferences and data I need in Mail etc. Actually it seemed like Time Machine did not copy all of my prefs for various apps onto the new install which has been kind of frustrating.