I found the issue with 2. (xxx.local extension as part of the wireless client name) seems to be as a result of the airport utility software upgrade. another MBP with version 6.1 shows my printer- wireless client name: "Canon-Printer ". On a MBP with airport utility version 6.3 it shows the name "Canon-Printer.local"
So that problem seems to be because of the airport untilty upgrade. now to get it downgraded back to version 6.2...
I can't speak to the .local issue; however, I had similar problems with connections after the 7.6.3 update. In my case, I have several devices, some of which are set up to use the 2.4 GHz band on the AEBS, while others use the 5 GHz band. I found that none all of the devices that were having connection issues after the update were attempting to use the 5 GHz band. When I switched them over to the 2.4 GHz band, they all started working perfectly.
You could try either disabling the 5 GHz band on the AEBS, or, if your 5 GHz band is named differently than your 2.4 GHz band, you can just switch to the lower band on your computer.
Also, for your Time Machine backup, maybe you want to do a permissions repair in Disk Utility. If your backup disk is connected directly to your computer, it shouldn't be having an issue like that. However, maybe the Airport Utility installation messed up some permissions... (just a WAG here...)
Had the same problem -- endless "looking for backup disk..." after updating firmware to 7.6.3, but solved it as follows.
While endlessly waiting, I noticed that if I tried to connect to the Time Capsule data folder through the Finder, it could see the volume, no problem. But then I tried to unmount the volume, and it gave an error, something like "cannot unmount, because it's already in use by the Finder."
The solution is to relaunch the Finder. I did this by rebooting my Macbook. But I suspect you could probably also solve it by doing command-option-tab, clicking Finder, and clicking Relaunch.
Hope this helps.