Can I use the Twig template engine?
Yes, you can, here is an article about my own experience with it, working with Bootstrap http://makina-corpus.com/blog/metier/2015/develop-with-twig-in-drupal-7

So http://www.drupal.org/project/tfd7 is the way to go if you want to write template files with Twig syntax, it must be moved to theme/engines. 
Note that development is now happening at https://github.com/TFD7/TFD7 and documentation is at http://tfd7.rocks.

Don't be confused with the project https://drupal.org/project/twig, unfortunately, it is only an input filter because modules cannot provide theme engines in Drupal 7 (due to license reasons it seems).