The Publish Transactions you are adding in your event handler are before the original Publish Transaction, because your event handler triggers on the “Initiated” phase. That is: before the original Publish action executes.
If you want to add them after, you should subscribe to the “Processed” phase.
Publication Key is a part of Publication metadata, it has no direct relation with the publishing content, so republishing of content wouldn't be required. For a DXA application, you probably need to republish Publish Settings page only.
AFIK it (changing) has no impact on import/export either.
However, do make sure to update any mappings in Topology Manager ...
I think the error message is confusing. Access to the tridion_cm database is failing, and because of that, the login for your User (WORKGROUP\EC2AMAZ-TUJ3HK0$) fails. This does not mean that the database connection makes use of your user's credentials, at least I hope not.
You should check the database credentials configured for the content manager. This ...