MacOS – Should I have the Mac on until “external” hard drive encryption process is done

You could use Apple's OS X Recovery Disk Assistant to create a bootable recovery disk on a USB thumb drive. Since you've backed up to an external disk using Time Machine you could replace the hard drive, boot to the recovery disk on your thumb drive, use the recovery disk to format the hard drive and then reinstall 10.8/Mountain Lion. Once Mountain Lion is installed you will walk through the same setup procedure that occurred when you first bought your Mac. One of the first questions in the setup process is if you would like to restore information from Time Machine, which allows you to connect your Time Machine disk and restore your backup.

If both the HDD and the SSD are running significantly faster when installed in the external enclosure rather than the internal bay, your SATA cable is likely the culprit. Both should run faster when connected via the internal SATA bus vs external USB 3.0 bus. You can run the Apple Hardware Test (which doesn't test your drive, but does test the connection), but it's prone to false negatives.

If you are close to an Apple Store, my advice would be to take it in for a free diagnostic (AppleCare won't cover the cost of replacing the cable since you already swapped out the drive, however). At any rate, you can get a definitive diagnosis and then decide whether to go through Apple, use a third-party service, or a DIY solution as posted in the link above. BOL.