Okay, onwards and sideways....
I thought I'd have a go, and add my username where I could be sure it should be. (There are lines in the Ubuntu group that are'nt in the Mint group.)
The good news; - I've now got my sound back.
The bad news; - sudo still doesn't work; I still don't become root in the terminal, and I still can't use Synaptic, although the message has now changed. It now reads; -
Failed to run /usr/sbin/synaptic '--hide-main-window' '--non-interactive' 'parent-window-id' '46137347' '--update-at-startup' as user root.
The underlying authorization mechanism (sudo) does not allow you to run this program. Contact the system administrator.
Possibly I need to add my username to another line in /etc/group.
I don't know if it'll help, but this is what it looks like at the moment; -
Can you tell if any lines are missing? Or should my username appear anywhere else?