The payouts are not based on daily or weekly rather it's based on once you reached the withdrawal limit set by the pool, for example, in F2Pool the withdrawal limit is 0.005BTC so as long as you received the amount in your account you can withdraw daily. Most common payout method is BTC but you can find other payout models based on the pools.

Yes, you can, but using an address created by an open source wallet is recommended.