Hi Thomas,
My biggest concern would be that you would somehow end up with 2 subscriptions under two different Adobe ID's. I don't think this would be the case but I don't really know. I did searches on your name, street address, and email addresses and didn't find any orders. Regardless, if this did happen we can fix it, its just in the interest of time and getting you running reliably at this point.
You won't have to reinstall any of the apps on the iMac,worse case scenerio, maybe sign out and sign back in again with the same Adobe ID you used on the laptop.
I think Ankush Rattan would most likely arrive at the same conclusion we are here, but it wouldn't hurt to wait if you can. He could walk you through the sign up process if needed.
-Dave