I think for most of the major exchanges like Coinbase, Bitstamp, Bitfinex, Kraken, etc you have to be verified somehow either by your bank or other documents, before you can actively trade.
If you want to trade with minimal verification probably BTC-E or Cryptsy will do.
I am fully verified with coinbase as in proof of ID and residence. Buy doing this I was able to buy and sell btc for euroes until last week. I realize if I wanted to sell for euroes and then transfer the euroes into my UK bank I would have needed to verify my UK bank account. This was not my plan however as I was going to buy back into btc using the euroes in my coinbase euro wallet. and then when I have say doubled my bitcoins using this method I was going to transfer the bitcoins back into my own bitcoin wallet
I am not an user of those big exchanges but as far as my experience goes with my local EU exchange I can trade without authentication as much as I am under certain threshold - for example: I can freely seling bitcoin until I reach 300 euro in transactions.
Yep that's correct.