When/where to include my own function library (like common.inc)? When, hence where, should I include my own library of functions that I will use throughout my whole project. 

Drupal has it's common.inc that is included in drupal_bootstrap during the fase DRUPAL_BOOTSTRAP_FULL. I also have a common.inc with functions that I use all the time, in different modules. I want to include this file once so I can use it everywhere without having to include it in each file separatlely.

So what is the right location/hook to do it?
Probably best to put it in it's own custom module, e.g 

sites/all/modules/custom/mymodule_common 


and enable it as required.

You can make it a dependency of other modules by adding it to those module's .info file, e.g

dependencies[] = mymodule_common


You could also use subfolder and standalone file in sites/all/libraries/common though the module route is recommended.