Can I use the effects of Kaiju cards my opponent gave to me?
You sure can! If you control a game object, be it a monster, spell or trap, then you can do with it everything you can do with a monster you summoned yourself. 

Note that this includes paying any costs that an effect requires - in your particular case, you'll need to remove kaiju counters from the field to activate Gameciel's effect. Note also that even though you're the controller of Gameciel, your opponent is still the owner, meaning that when it gets destroyed or otherwise sent to the graveyard, it'll go to their graveyard, not yours, or if an effect tells you to return Gameciel to the hand, it'll go to their hand as well.