For remittance, invoming_invoices and statements: I would create projects and assign respective aliases to these projects (remittance, incoming_invoices, statements). Configure the project so that incoming emails referring to this project's alias create issues, not tasks (issues are better to handle communication with external people).

Then define the steps in your projects according to your organization. As an example, for remittance, you can use the following steps: New, Accepted (when you get the valid document), Processed.

For orders@mycompany.com, you should create a sales team and assign it this alias. This will create leads that you can convert to opportunities and sale orders.