If you have 0 cards in your hand after using Final Destruction and the turn ends, your opponent proceeds to resolve any end of turn effects, and begin their turn as normal.
Using Different Strokes will not protect your opponent's SIGNI when your opponent downs their LRIG when they pay the cost of the ability of this spell, as this spell does not affect your opponent's SIGNI directly when it gives your LRIG an ability.
Your LRIG's ability will affect your opponent's SIGNI. Therefore, it can be defended against by Contempora, Inexplicable Superboast's second ability.
Your LRIG's ability will affect your opponent's SIGNI. Therefore, even if your opponent has Arcgain, Archangel of Pioneering on their field, Arcgain will still be vanished by your LRIG's .
Arcgain's ability will protect against your opponent's spells, but not protect against the effects of your opponent's LRIG, including the ability that this spell will give your LRIG.