"Twig_Error_Loader: Template "file.html.twig" is not defined" for new twig template file
Cache is a pretty broad concept, and sometimes mean content caching and sometimes they mean broader things like config settings and the theme registry. Disabling content caching is still a bit confusing to me so I can't help with that part of the question.
But the theme registry has some pretty simple rules. If you add new template file (or change it's name or move it to a different folder) or if you add a new theme callback (in hook_theme) or override to a theme callback (hook_preprocess...) then you MUST clear the theme registry before they will be active. To do that, the simplest method (without additional modules or drush) is to 'Clear all caches' under admin->config->development->performance.
Of note, sometimes the theme registry is referred to as the theme cache. While I know it's probably possible, I don't think that disabling the theme registry from caching is a standard practice in development.