I forgot one thing. Yesterday I closed the access to one file and I found that all of folders in HD and their files were closed, too. I couldn't find where this was possible, though I found a consol command to fix it here. When I went to close the file in Get Info there were arrows pointing left and right that would change the file to other ones. I thought that odd enough to mention it, hoping one of you will find it a clue.
I was able to reestablsh contact with TC disk's sparsebundle by connecting the iMac to one of Ethernet connections of TC, the modem to the WAN port, and a factory re-set, setting Airport Utility to Ethernet without Airport successfully. (Airport stopped working somewhere along the line.) The TC was now steady green. I used used external HDs from my computer and old, failed TC for backups.
I did some reinstalling and reworking of the two OS 10.6.8 on the iMac on of two partitions this morning, but I saw at some point that sparsebundle was lost. TC was still green, but no joy. I went through it again with Airport Utility, TC glows green and reports success, but no sparsebundle. Then I hooked up an external backup but found that I had lost conact with the external HDs that I do backups on. The only way I can use the internet is to connect it directly to the modem. When TC went missing Airport still worked and with the iMac connected to one of Ethernet ports of TC, it connected.
Ah, so! I'll look for an item I found in the Community connection and see whether it rings a bell.
This I think is the final. At one point (above) the TC was in fact sending WiFi, my Kindle picked it up. I didn't recognize the source and Airport didn't either. I'm in an apartment complex and I didn't think much of it--but I should have. I was able to reset TC(above) and get to go Green, but Kindle found no useable WiFi signal. Green, but no signal. I was never again able to connect with sparse.bundle hooking up Cox's modem to TC and the TC by Ethernet to the iMac. I was able connect to the internet, but nothing else: TC steady green. I think its HD may be dead, no starting vibrations felt--I'll get to the Apple Store and see. It's in warrantee.
I downloaded a trial antivirus or cleaning program and it found a non-native "helper" in Unix that sent out error messages that stopped its going on. Reading David Pogue's The Missing Manual, I learned that a guest account protected-file in the Unix sub-system is beyond an administrtor's reach. If he tries to get in he'll get, "Permission denied." This helper goes well beyond that. He takes preemptive action before he can be asked. I know this beast! I would have to say, in a way, its solved: I'll take the iMac back to the authorized repair firm and suggest they fix it. Transportation will have to be arranged. I am waiting for an iMac I think will be revealed next week. I'll have to not connect it to anything that's been connected to my old one: the bug travels. All of the files and folders in old backups are now closed with altered permissions. When I go to have the TC evaluated and maybe replaced, I'll ask them what to do.