Yes, I am pretty sure they are downloaded and installed on the EHD, Drac.
Edit: The PC shop has just rung me to say they need me to bring the EHD back into their shop - they said they can copy my games but I told them it was not that simple with BFg games.
I think i had better now re-download all the games on this new hard drive onto the old Seagate so that I can still play them. Thank goodness I dont have to pay per GB for my downloads!
I'm pretty sure that Big Fish shortcuts will be pointed at the external drive (same drive letter as the one where they were installed) and not at your internal hard drive. So if everything is transferred to a new external hard drive and the file structure is preserved, the games should work fine.
They may play from your hard drive too if you adjust the path of the game shortcuts. I don't think the activation information is kept in the game folder. It's probably hidden in your registry somewhere. So I don't think it's necessary to download all the games from Big Fish again -- which would take forever.