Does copying a spell on the stack count as casting it?
No, the copy is not cast, and would not trigger Paradox Engine.

From the Gatherer Rulings on League Guildmage:


  10/5/2018     The copy is created on the stack, so it’s not “cast.” Abilities that trigger when a player casts a spell won’t trigger.


Paradox engine cares only if you cast the spell, and a copy of a spell made by League Guildmage is not cast.