Hello,
Seems you created another account to try and know if the bids you get on the same project will differ, but I will again like to make the same offer once more.
I know you have listed your budget as $250~ but a app with native features like that, let alone a game cannot be developed in that budget, that also on 2 mobile platforms.
If you can increase your budget to $500~, I can create this quiz game on crossplatforms (ios-android) for you.
Furthermore, like my past game, I will create the quiz questions, answers and categories part of the game Dynamic, that way you can update questions-answers and add more topics to the quiz without requiring modifications to the game itself and requiring users to update the game.
I can create a API for your game and topics, via which you can add more topics and quiz Q-A to your game from a remotely hosted JSON file, which can be loaded into the app to add more content by a simple tap of button.
PS: I can do this project on both platforms in $250 if but only for top-level categories (no sub-cats), will include leaderboard but no social network login. Not to mention that Friend to friend aka Multiplayer battle system can't be done under $750~ budget, you are asking a multiplayer game on 2 platforms.