Skype 7.10 application no longer able to log in any user, check for updates, or change online status

Question

Hello Community,
Thanks in advance for looking into my problem report.
First off, I'm running Skype version 7.10 (777) for Mac, on my MacBook Pro Retina running Yosemite 10.10.4.  I've been using Skype on this setup flawlessly for months.  Recently, after adding some more credit in advance of making an international call (which worked fine), I ran into a weird problem.  I can't be more specific about the initial problem, as I kind of dismissed it at the time, but it was some kind of error window from Skype regarding my attempt to add credit (which did work, btw - I see the updated balance and my bank account was debited as intended).
Now however, I cannot use the Skype application in any way.  When I attempt to sign in - whether on my account (with the right password or a bogus password), or on another valid account, the application just displays the spinning balls graphic for several minutes and then reverts to the sign in screen.  After rebooting my Mac, I was able to use Skype again, but now it's back to this behavior after I quit Skype.  I've got a lot of work-in-progress right now and don't want to keep rebooting to try to get past this.
Note that my account/credentials is clearly not the issue; I can use Skype from my iPhone 5S no problem, and from Skype web on the same Mac.  But the application itself on my Mac is DOA, whether using my account or another person's.  Even the Check for Updates just stalls forever.
I have already downloaded the Skype app again and paved over the old one, and that makes no difference.
I also trashed my Library/Application Support/Skype, and restored a copy from backup prior to this problem, but it made no difference.
I've verified that my Mac firewall is not the issue - I've tried it with the firewall on and off without affect.
If anyone has a clever idea - or knows how I can get some kind of diagnostic log about what the app is doing when it's trying to contact the mother ship - please let me know!
Thanks again
[Topic title updated by moderator to include version number.]

Answer 1

I'll add that when I run devenv.exe with the flag /TFSLink, VS opens a Team Explorer window that contains the following text:
An exception was encountered while constructing the content of this frame. This information is also logged in "C:\Users\<username>\AppData\Roaming\Microsoft\VisualStudio\11.0\ActivityLog.xml".Exception details:System.NullReferenceException: Object reference not set to an instance of an object.   at Microsoft.VisualStudio.Shell.Interop.IVsShell5.LoadPackageWithContext(Guid& packageGuid, Int32 reason, Guid& context)   at Microsoft.VisualStudio.Platform.WindowManagement.WindowFrame.GetPackage()   at Microsoft.VisualStudio.Platform.WindowManagement.WindowFrame.ConstructContent()