Can I automatically replace the "civicrm" from all URLs with another string? (drupal7 + civicrm 4.7.4)
There is no easy way to do this.

You might be able to create a drupal module that is able to get the myOrg urls and then renders the CiviCRM version of it.

But then you still have the problem that the internal links that CiviCRM creates will be in the /civicrm/* form.

To solve that you could first get the CiviCRM page as a string in the drupal module, then do a search/replace and output it.

Very hackish, probably not possible to get it 100% right. I would advise not to.