If I mulligan and start with a {gemstone caverns} in my hand can its ability still activate?
Yes. Your "opening hand" is the cards in your hand of cards you decide to start the game with, after any mulligans.
RESOLVED
Great thanks
Also can you wait to activate the ability before the opponents end phase?
No. It says you may begin the game with it not play it during the opponent's turn.
Yup that sounds about right thanks and happy holidays