How to check the modules installed date? How to check which modules was installed on what date?  I am using Drupal 6. I can not find any answer in Google.
You can't have that information with default core and contrib module.

You can get the time module upload on your server by checking last modify time of module file.

If you realy want to have module install information, you should create a patch that change the module_enable function on includes/module.inc file.