It could be something that is resolved with a SMU reset, so be sure to try that one as well, which is the document I linked to before, since your iMac turned out to be a G5.
You will be able to do a complete system restore from the Time Machine backup, if you have the Leopard installation disc that you can use for start up. The command to restore the complete system is in the menu bar of Installer, under Utilities.
However, I would not recommend doing so (restoring the complete system), if the Mac being targeted is not the same Mac (or at least the same model Mac). You should be able to use the Time Machine backup as the "data source" with Migration Assistant. In that case, the eMac would already be set up with it's system and you would be migrating in your user data (and maybe applications). This would work best if the eMac was also running Leopard (10.5.8).
You can also access your data in the Time Machine backup folder directly, using Finder. So, you could even "manually" access and restore your data from the Time Machine backup to a different Mac.
If the external drive is USB, hopefully your eMac is a model that has USB 2.0. Actually, "hopefully" your iMac G5 will be working fine... 🙂