How to export Panels Everywhere template code to a module
You can try features module.

The features module enables the capture and management of features in Drupal. A feature is a collection of Drupal entities which taken together satisfy a certain use-case.
Features provides a UI and API for taking different site building components from modules with exportables and bundling them together in a single feature module. A feature module is like any other Drupal module except that it declares its components (e.g. views, contexts, CCK fields, etc.) in its .info file so that it can be checked, updated, or reverted programmatically.

the features module converts every entity (views, content types, ...) in to a module and it can easily be installed and modified every where.