Is there a maximum amount you can bet on the dice game and what is the maximum Freebitco.in would pay out, if people win really big amounts?
You can bet any amount. But the profit you make in single bet cannot exceed 20 BTC.

What is the biggest payout that Freebitco.in have ever paid out, apart from the Lambo draw winnings in BTC? ( Is the payouts automatic or do you flag certain withdrawals at a specific amount for manual payouts? )
Given the huge profits they make, they should have a big bankroll. So, I don't think there's any limitation on withdrawal amount other than the minimum withdrawal amount.