Time Machine keeps failing

Question

Hello,
This is the most annoying problem I've run into in a while. Somewhat discouragingly, I've found a couple of other posts of similar problems — and none have answers. But let's try it.
Previously, I had the following setup, and it was working reliably: My iMac is my main system. I had a 2TB WD FW800 drive attached, using an adapter out of my T'bolt port. The iMac is cable-connected to my router (sadly an AT&T router, but that's another issue entirely). My Mini is used mostly as a media center, and it is also cable-connected to the router. The drive was partitioned into two partitions, both shared on my home LAN. I used one of the partitions as my Time Machine volume, and the other as utility storage. Both computers used that volume for TM; the iMac used a Backups.backupdb folder, the mini used a .sparsebundle. As I said, it was operating reliably.
Then I knocked the WD drive off my shelf. It no longer powered up. I decided to just replace it. I purchased a G-Drive 2TB (FW800), connected it, and attempted to configure it as I had the WD, i.e., with two partitions, and use the larger for TM. What seems to be happening is that after some fooling around that I feel I oughtn't need to do, I'll get one apparently-successful full backup on the iMac. Then when a subsequent backup is attempted (either waiting for one to start by itself, or invoking "Back Up Now"), it fails with one of those generic "Unable to complete backup. An error occurred while creating the backup folder".
The mini has similar issues. Often (not always) the backup fails with "the network backup disk doesn't support the required AFP features". When I check the permissions of the external drive, the owner has been changed from "me" and "staff" (or maybe "admin") to "system" and "wheel" (never understood that one). If I add myself and admin to the permission, give them r/w privileges, I'll get one apparently-successful full backup, as with the iMac. Then, when invoking "Back Up Now" from the TM menu, it fails. (Actually now, I'm not getting a failure message, but nothing seems to be happening; the "latest backup" is still the original. It goes thru "Preparing backup..." but never copies anything.)
I have tried reformatting and repartitioning the drive (GUID), I've tried using only one partition, I've reinstalled the 10.9.4 combo updater (on both computers). I also note that if I select the current TM Backup and click "Remove Disk", it doesn't reappear in the list below (ought it?), so I've tried using the other partition for my TM volume, I've tried new volume names. Sometimes after various flailing about TM will ask me for logon credentials, sometime not. Sometimes reformatting is a challenge, it won't let me unmount the disk unless I turn off file sharing — again, something I didn't used to have to do, I believe, and even then that doesn't always allow me to dismount it.
I have the feeling that by now everything is really screwed up. How might I totally reset Time Machine, as if I had never ever used it at all (and sharing, for that matter)? Is it even possible? Looking at another approach, is "Archive and Install" of OSX still an option is these disk-free days? Is there something I can do or examine regarding the drive? Disk First Aid finds nothing (although it did find problems at one point, then, feeling hopeful, I reformatted and started over, but the problems still recurred).
I just tried reformatting and starting again, but I didn't see anyplace where I might do anything different. The mini gave me the "AFP Features" error as above; I added myself and administrator to the permissions list and gave them r/w privileges, after which I got the "your backup disk has changed identity, do you still want to use it" message. I clicked "Use Disk", and it seems to be proceeding. Both systems are now going thru a first backup. Problem is it'll be hours before I get any results. The iMac is going thru an apparently-normal first backup.
Any advice would be helpful. I can't even figure out how I might present this to the Genius Bar or a One-on-One session.
Jeff

Answer 1

OK, a little bit more...
If, on the mini, in TM prefs, I click Select Disk... my TM volume appears twice, once at the tom, the current disk, and once in the list below. However, when you hover the pointer over each of these entries, it shows the URL; the "current" disk displays afp://Andomeda._afpovertcp_._tcp.local/Aphrodite but the one in the "choose me" list displays afp://Andomeda.local/Aphrodite . Now, if I Remove the current disk (the afpovertcp volume), and Use the other one, the one that then shows at the top, the current one, is again afp://Andomeda._afpovertcp_._tcp.local/Aphrodite .
So now I'm completely confused.