Did you clean install Mavericks?
If you have recovered your files and are happy that everything is working fine.. Erase the TC and start over. there is often a problem with the TC and TM getting themselves in a knot.. especially since Mavericks came out.
Most important.. do you have IPv6 set to link-local for whatever connection you use.. wireless or ethernet.. set it via the network preferences in the computer.. although it is required, Apple did not preset it correctly.. Automatic is not good enough.
Can you pickup the TC in airport utility? If not TM certainly won't pick it up.
I also recommend you redo the naming.. Mavericks is very tight with names.. you need all names to be short, ie 2-10 characters preferred... no more than 20. No spaces and no special characters like apostrophe.. pure alphanumeric.
Apple again do not help themselves allowing you to make illegal names via the utility.
Reset the TC to factory.
Unplug your TC/AE. Hold in reset. and power the TC/AE back on.. all without releasing reset and keep holding in for about 10sec.
Release it when the status light flashes rapidly. If it doesn’t flash rapidly you have missed it and try again.
Be Gentle! Feel the switch click on. It has a positive feel.. add no more pressure after that.
TC/AE will reboot after a couple of minutes with default factory settings and will wipe out previous configurations.
No files are deleted on the hard disk.. No reset of the TC deletes files.. to do that you use erase from the airport utility.
This will eliminate your password problem.. the password if it is requested is public.
Erase files on the TC.
Open the airport utility and go to the disk tab.
Select erase disk.
Quick erase is enough.
If you click the window you will see the following erase options.
Zero Out Data is the next option.. that is a one pass low level format.. it takes too long even for one pass is several hours.. as for 35 passes I would say about a month.
Quick erase simply deletes the file table.. it is not secure for selling the TC but works find to simply start afresh with the TC.
Apple have made ipv6 necessary to the function of the TC.. even if you use ipv4.
Simply go to the network preference in your computer.. open the advanced settings for wireless and ethernet (if you also plug in).