It's interesting that even though the first INSERT statement is not part of an explicit transaction, the ROLLBACK inside the trigger negates the INSERT. If there was no trigger...

Yes,

http://msdn.microsoft.com/en-us/library/ms189292.aspx#BKMK_persisted states that:

"You can create an index on a computed column that is defined with a deterministic, but imprecise, expression if the column is marked PERSISTED in the...