Possibility#1

If a trigger is not firing, then the insert/update/delete of the SObject is failing before the trigger is being invoked. For example, an after insert/update trigger will not fire if there are validation rule errors during the insert/update.  Or, if the before insert/update triggers execute an add.error to prevent the DML from committing.

Given that it works in stage but not PROD, you need to verify that Stage mirrors PROD in terms of validation rules and workflow field updates.

Possibility#2

Another thing to check is that if the stage testmethod uses @isTest(SeeAllData=true) it is highly likely to have different behavior in PROD than in Stage where the dataset size will be different.

Debug logs

As for debug log levels, start off by just running with Validation and Database logging and turn off Apex code and System logging to see why Possibility #1 (the DML operation is not committing) thus preventing the trigger from executing. 

If you need to see Apex code logs and you're exceeding max debug log, see this post for suggestions http://salesforce.stackexchange.com/a/11251/2602

How to view the detailed debug log of the change set validation in production
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