Where to look for clues as to why SQL Server become unresponsive/crashed?
Only administrators may connect at this time.


Sounds like the UI has issued ALTER DATABASE ... SET RESTRICTED_USER and then hung. This hasn't actually hung SQL Server, it has only prevented non-admin users from connecting. You could get around this by making your application use sa - I'm kidding.

Honestly, stop using the UI for log backups - it's not very good at things like this, and it's hard to pick all the right options for success. BACKUP LOG ... TO DISK is much more reliable, transparent, error-free, and even source controllable...