Does Drupal strip out style and span tags? I was trying use span and style tags in pages, rather than in CSS and it seems that Drupal is configured to strip them out whether the input format is Filtered HTML, i.e. they are applied to Full HTML.

Is that something which can be fixed only by editing the core files, or are the some modules or settings in Drupal's admin settings that can override that?
Go to Configuration -> Content Authoring -> Text formats. Here you can configure what tags are allowed.

All unallowed tags are stripped out,  is always stripped out.

Check out https://www.drupal.org/project/wysiwyg_filter for more advanced filtering.