If things were running fine the day before and all of a sudden ALL of your tasks are failing on the distribution portion, then I would think the issue would be environmental, not a QlikView error, per se.
The error in the log provided:
Error Failed to connect to QlikView Server. This error is most likely because the server is not contactable or that the account the QlikView Distribution Service is running as is lacking permissions. Address=QlikViewStorage:4747, Exception=System.Net.Sockets.SocketException (0x80004005): An attempt was made to access a socket in a way forbidden by its access permissions 10.95.130.91:4747
points to potential network issue. Have you confirmed port 4747 is open? When doing a server distribution, the traffic is over the QVP protocol on port 4747. If you configure the task to use a folder distribution instead of server distribution, does it still error out?
Hope this helps