Pendant of Prosperity's effect doesn't count as playing a land. It can put a land card onto the battlefield even if you've already played your land for the turn, and even if it's not your turn.

In a multiplayer game, if Pendant of Prosperity's owner leaves the game, Pendant of Prosperity leaves the game with them. If Pendant of Prosperity's controller leaves the game, Pendant of Prosperity is exiled (assuming that player wasn't also Pendant of Prosperity's owner).

If you control your own Pendant of Prosperity and activate its last ability, you'll draw a card, then you may put a land card onto the battlefield, then you'll draw a second card, then you may once again put a land card onto the battlefield.