The only real restrictions on origami are no cutting, and models requiring multiple sheets are expected to have locking mechanisms which make it possible to keep them together without glue (though glue is not forbidden and is encouraged to be used in shaping). Paper doesn't even need to be square, it's just desirable. Curves and twists are completely acceptable, and curved origami is its own niche little area rarely explored due to the difficulty of making a clean curved fold. Jun Mitani is a master of this style.

It is still origami because there is no such thing as purely clean folds. You opinion is based upon a semantic definition rather than a functional one. Paper might look 2 dimensional from a distance, but it is a 3 dimensional material.

Even the simplest origami models create layers of paper where nothing is truly flat. Paper itself has a tendency to flex and buckle at the creases. Many folders learn to adapt to the variance introduced by paper thickness.

What you have labeled as aggressive twists in complex models are not achieved by such a blunt method. They are flaps where numerous folds have converged. Like the accordion folds in a paper fan, there are many layers that add bulk to the appendage. You cannot haphazardly twist this to produce the final rounded shape. The creases concentrated in the area make it more prone to tearing.