Sounds like a mess! So you booted off another hard disk and/or computer using target disk mode. Did you try using it for awhile like this or immediately attempt to copy/upgrade to OS. I am under the impression you cannot simply copy/paste the OS in the Finder because this does not copy some of the underlying unix files that are required. I would think that if the eMac would run fine for a somewhat extended period of time via target disk mode it would imply that the logic board is ok. When you are in target disk mode and you open Apple's Disk Utility does the S.M.A.R.T. status of the eMac's hard disk look good?
All this being said, I purchased an iMac G5 about a year ago that had some frustrating issues similar to what you are describing and the more I try to use common sense to narrow down the problem the more things became confusing. In this case it was the logic board, and once it was swapped out things returned to normal. I could not reliably boot off another computer in target disk mode, so that is the reason for my line of thinking.