Anyone knows when one starts earning double interest from keeping coins in Sharex as opposed to staking from wallet? I would transfer coins to Sharex to earn more interest but I don't know when it starts. Anyone?

Not yet, staking will start when the initial FeeShare sale closed and will stop when no ShareCoin is left in the exchange's wallet.
And FeeShare sale will open again to generate ShareCoin for the exchange's wallet, and then when the sale closed, it will start paying interest to ShareCoin's owner, and stop when no ShareCoin left, and the FeeShare sale will open again to generate ShareCoin for the exchange's wallet, and then when the sale closed, it will start paying interest to ShareCoin's owner and will stop when no ShareCoin left and will open again,etc..... this keep on going until 50,000 FeeShare  are distributed to the community.