There appears to be so many exchanges / middle-mans on a bitcoin market (e.g coinjar.com) which can help to buy / sell bitcoin for a cut (usually 2-4%). Do they simply trade bitcoins using their own bitcoin wallet and just keep the client's balance of cash/bitcoins in their system?
They don't sell nor buy anything they are in the middle of a transaction. Check this out, Active buy orders are people who want to buy, and active sell orders are people who want to sell. The exchange is the place where all that people meet.
If you have Bitcoins and want to sell them for real money you have two options, you can sell it to the first buy order for the specified price and the transaction will be finished in that exact moment (if the buy order is for the exact or more amount of Bitcoins that you are selling) and you will be credited with real money that you can withdraw to your bank. Or you can set your own price and wait to see if someone wants to buy it for that price. Depending on the price you set it may take little time, longer or forever.
For every acount a wallet is created where you transfer your Bitcoins, real money is deposited in real banks and transfered to and from those banks.
But I am more familiar with Coinbase.com, there you can buy/sell/send/request bitcoins.
It is set to the local currency of your country, and you will have a wallet automatically. The wallet address is found in the account settings/bitcoin addresses. The website automatically creates a bitcoin wallet address every 9 days. Don't worry about that, you can use any or the same bitcoin wallet address for forever, so it doesn't matter if the website keeps creating a new address, the address that you use to receive is reusable.
Goodluck. If the site like you mentioned, coinjar.com is a bitcoin exchanger, yes, they do have a wallet system where you keep your bitcoin balance. The good thing about that is that you don't have to backup your wallet offline because the site backups wallet for you. Like in coinbase.com they backup everyone's wallet offline and update the balance everytime you make a transaction.
Hope this helps.
You could buy or sell a physical bitcoin like a casacious coin or BitBill (or even a funded gift wallet like aLittleBitOfCoin). That kind of exchange does not even require computer access. You could exchange offline paper wallets but then there is more reason to question the secrecy of the private key.
No, in a strict sense. The first bitcoin transaction predated the first wallet by several months, and there's no reason that couldn't have been a p2p cash for btc transaction. The first transaction was with Hal Finney in January 2009 (the 10 BTC test), and the first wallet was, I think, with v0.3 of bitcoin, sometime later in 2009.