Casting a spell is a keyword action that involves placing a card (or copy thereof) on the stack, making choices, optionally activating mana abilities, and paying its cost.
Activating an activated ability (such as Prototype Portal's second abiliy) is not casting a spell.
Creating a copy of an object is not casting a spell (even if the object being copied is a spell).
Putting an artifact on the battlefield is not casting a spell.
No. Paradox Engine cares specifically about casting spells, whereas Prototype Portal has an activated ability. These are different things (though they can have the same result, e.g. an artefact entering the battlefield).