plain vanilla network - cable modem, TC as router, extended with airport express, time machine looking for disk wirelessly. Although this new issue with no Internet access and not being recognized is a problem for both wireless and wired.
On extended network.. the other thread strongly suggests too many issues with 7.6.3.. some people seem to resolve it by jumping through a series of hoops but it appears to offer almost nothing useful.
https://discussions.apple.com/thread/4787229?tstart=0
So go back to 7.6.1 (if not to 7.5.2 which IMHO is even better).
To help with the problem .. AFAIK there is no work around yet.. and no bug fix in the latest firmware.. rather additional new ones. But these might help things work better.
On the TC.. Factory reset to get rid of all settings. Start over.
I would strongly recommend this.. at least. Factory reset does not do anything to data on the disk.
Set all SMB names. ie TC name.. eg TC
wireless 2.4ghz.. TC24ghz
wireless 5ghz TC5ghz (use different name to control what is doing what).
Hard disk share name. Data (some TC use names with spaces.. don't!!)
Fix the wireless channels. For 2.4ghz there are only 3 non-overlapping.. 1, 6, 11.. test them in reverse order as 11 or 13 in ETSI regions are less used.
Use only WPA2 Personal security. Especially for extended wireless. Use password 8-12 character pure alphanumeric.
I recommend you set dhcp lease time shorter.. 20min will not increase network traffic to any significant degree but keeps everything updated.

Default is one day.. things get lost over that time with computers and network shutting down, sleeping etc.
Once you have done the TC setup then do the Express to extend again.
Tell us how it goes.