Are all trades stored in block chain? Are all trades stored in block chain?

Are all trades, like mtgox exchange, gambling, etc. stored in block chain? Or, few trades could have been skipped / ignored? 

Consider this case, 'a' transfers 10btc to 'b', and then 'b' transfers 10btc to 'c'. Is it possible to store this as single trade, as 'a' transfers 10btc to 'c'?

Is it possible for an exchange to include fake bid/ask orders for it's own benefit?

Is it really possible to differentiate between two transfers, where in one transfer is between single user accounts and another one is between two users?
Within an exchange's ledger system, value may be moved between accounts without creating actual transactions in the public Bitcoin ledger, the blockchain. Exchanges that use this method keep all of their reserves in one set of accounts and simply debit and credit user accounts as necessary. Only when a Bitcoin transaction is necessary to move value from one account to another outside of the system will a Bitcoin transaction be created and broadcast.

In other systems, an exchange or other entity may actually create a new Bitcoin transaction for every trade. This method is more inline with the ideals of Bitcoin, but costs more because of the transaction fees necessary to expedite/facilitate processing.