You can see the list of those by executing the command "unspent_outputs" in the wallet. Each output has some "amount" of coins. When sending coins, the wallet will pick some unspent outputs, enough to cover the desired transfer sum plus transaction fees, make them inputs for a transaction and send whatever is above desired sum back to your wallet as change.

While this basically just means that new coins need to stay roughly a day in your wallet to start competing for block reward, there is one more caveat. After being able to mint a block, the output will be recycled into another output of the same amount. Additional output with the block reward will also be received by the wallet. All those new outputs also need to mature for 800 blocks. This means that really big outputs will produce less rewards than their fair share. For example if you have output of 100000 coins and the sum of all staked coins is 20 million, this particular output is expected to get one block in 200. However due to maturity requirement it will get on average one in 1000. This is however solvable by just splitting your outputs into smaller ones (covered later). For amounts of 1000 and lower the maturity requirement will not have measurable impact.

That is output of 2.89 coins has the same chance of earning a block reward as output of 2 coins.

However in order to make larger outputs more likely to earn rewards, the amount of coins in the coinstake transaction (that is the amount of coins in the single output that is the input in the transaction) is made public. That for example allows outside observer to bet that two blocks that are more than 800 blocks apart and have some specific number of coins (for example 4563.8943100131) in coinstake transaction belong to the same person/wallet. However if the amount of coins is less specific - say 100.0 and many blocks have coinstake transaction of 100.0 coins then practically no information is revealed.

All you need to do is execute "start_staking" in the cli wallet, enter your wallet password and leave the wallet window open. You need to have daemon and the wallet running in order to earn stake rewards. You can do that on a remote Linux server by running inside a "screen" session.

However the difficulty algorithm will need time to converge to that value. So it is expected that if sum of staked coins is significantly less than the Proof of Work difficulty at fork time - blocks will come slowly at first. The vice-versa is also true.