If I have Prototype Portal copy an artifact but I have Paradox Engine, would Paradox Engine’s abilities activate every time I copy?

No.

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).

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    Technically, casting a spell can't place an artifact on the battlefield. Resolving a spell can. – ikegami Nov 6 at 12:47
  • (And please use [mtg:Card Name] to link to cards instead of using a link.) – ikegami Nov 6 at 12:48
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    @Ikegami Technically, if there were a spell that has placing an artifact on the battle field as a cost, then casting the spell would place an artifact on the battlefield. – Acccumulation Nov 6 at 16:14
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    @ikegami [mtg:Card Name] is fine as long as the card name isn't also part of other, longer card names. I personally prefer going to Gatherer, searching for the card, then use [Card Name](http://gatherer.wizards.com/whatever the full URL is, I'm just copy-pasting it). It's not much slower, but it's more reliable. – Arthur Nov 6 at 19:10
  • @Arthur Doing it that way makes it annoying when people click on the link though; instead of getting a new window popup with the card; you get redirected to the gatherer site. – GendoIkari Nov 6 at 20:46

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