Is there any module that can automatically convert a module to be used on a different version of Drupal?
I will not say that it is fully automatic, but Coder Upgrade, which is a sub-module of Coder, will do a lot of the grunt work for you.  However, it is not a 100 % solution, there are always some loose ends to tie up.  It is a good idea to use Coder Review afterwards to highlight the areas that need special attention. After that, it should be a simple case of installing the Drupal 7 version of the module, test it, and fix the remaining nits "by hand".

You can download and install it as part of your staging site or use the web-based version: http://dupgrade.com/

It is one way: Drupal 6 to Drupal 7.  I know of no tool that goes in the other direction.

See also: https://drupal.org/node/224333 for an overview of the changes from Drupal 6 to Drupal 7.