You are mixing up things.. the airport utility is 6.3.2
The firmware of the TC is 7.7.3
When you load the firmware it will simply restart and you will not see anything in particular.
Look at the summary page in airport utility and tell me what the firmware is. It is marked version. Sorry the edit is marked but the version is above that.
To fix your problems.. I strongly recommend a new name for the TC and its wireless.. names that are short, no spaces and pure alphanumeric.
Mine is TCgen4 .. you can make yours TCgen5 and use TCwifi for the wireless name. These names are helpful for Mavericks which is now using SMB (windows) network protocol not the previous Apple File Protocol.. AFP which is not as strict.
I also recommend you do manually mount the TC.. but check how the above works first.
Isolate out the TC..
Reset it to factory.. this is most important.
The Factory Reset universal
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. The time is not important.. it is the front LED rapid flashing that indicates you are in factory mode.
If it doesn’t flash rapidly you have released reset at some point 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.
Plug it into computer by ethernet.. LAN TC to computer.
Do an entirely manual setup.
No luck the TC is dog meat.. take to Apple for replacement.