How to template a view and be able to manipulate the content with it?
You should create new display modes for the content in question, and provide twig templates for them (e.g. node--type--display-mode; I am unsure of the pattern). Then you can change the View to show the rendered entity, and select that display mode, which will use the twig template for it.

This will be a lot easier IMO for what you are trying to do and make the View itself more maintainable long term. For example, maybe you wind up having to make multiple views... editing all those fields via Views is going to become a real pain real fast. Changing the field configuration in the display mode settings of the respective entity type will be reflected in the View immediately without changing the View itself (unless caching is enabled, then clear it).

See: http://www.mediacurrent.com/blog/drupal-8-theming-without-panels for some background.