Your confusing me now..
To add, if I connect the wifi to the TC wifi, it works. If I connect the wifi to my adsl router, it does not. However, if I have the wifi connected to the adsl router, I can see the TC in Airport, as working fine.
But in no case does the TM backup work?? Does the TM work if you connect to the adsl router wifi instead of the TC wifi?
So you have the adsl modem as your main router. The TC should be in bridge.
I have a few tasks for you.. none of which is guaranteed to work but we should be able to fix it.
1. Downgrade the firmware.. you can try the following steps without doing this.. but 7.6.4 is not helping.
In v6 utility in Lion you can simply select the TC, option click on version.. and all the older firmware will appear. Select 7.6.1 and downgrade.
2. Factory reset.
The Factory Reset TC Gen1-4. Or AE Gen1-5
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.
3. Redo the setup.
Use all SMB type names.. eg.. TCGen4 as TC name.. it is short, no spaces and pure alphanumeric.
Use TC24ghz for 2.4ghz wireless. Use TC5ghz for 5ghz wireless.. yes, use different names at least until you work out the issues.
Use WPA2 Personal security with a 10-20 character full alphanumeric password.
4. Delete the current TM setup and start again.
See A4 here. http://pondini.org/TM/Troubleshooting.html
So you will select the newly named TC but the TM should find your old backups and continue to use them.. only Mavericks it seems is too stupid to do this.
5. See if TM now works by wireless.