Hello!
You should never edit files in the folder skins\default\en\css - the first upgrade will ruin your work.
To add custom styles either use
Theme Tweaker, or
create a custom module (most likely you need a
theme module).
You don't need to run the classes cache re-building process after editing a css file in your custom module (unless you enable website optimization techniques like css and js aggregation).
The order in which css files are loaded depends on:
- name of the module added the file (alphabetically)
- whether there are other modules which that module depends on
- whether it is a theme module, or not
Please could you clarify why the inspector doesn't help? Is the CSS aggregation setting disabled on the Store settings -> Look and Feel -> Performance page?