For your mail question try this Pondini FAQ
It has specific section for mail.
For your two macs one works one doesn't.. welcome to the completely obscure way Apple systems now work.
Do a verify of both the source disk and sparsebundle in the TC..
If there is no issues.. load the widget A1 into the computer that has issues so you can get a full log file of what is going on.
You may find it is easier to just admit that TM no longer works particularly well with ML to TC.. there are too many bugs.. get an external drive and try that.. if it still gives issues.. I would recommend a clean install of the OS.. not an upgrade.
The most recent firmware upgrade merely added to the problems so if you upgraded.. and the problems started . then downgrade again.. you are now a beta tester even if you didn't know it.