I`m sure downloading the blockchain actually helps in confirmations.
No. It does not.
I think that part of the FAQ is about the Multibit client. Multibit does not require you to download the blockchain. Bitcoin-Qt does require the full blockchain.
Correct. Different wallet implementations have different requirements. MultiBit is a Simpified Payment Verification (SPV) wallet. Bitcoin-Qt is a full node.
Can you explain "doublespends" ?
That is where you get to spend the same money twice with different people. It was a common problem with digital money before Bitcoin solved it.
It's still possible to double-spend. However, it's much harder to do so, since the miners will not accept it. Unless you control the miner of course, then you could theoretically make it go through.
People don't always mean the same thing when they use the words "double-spend". It is not possible for two transactions that spend the exact same unspent outputs to both be confirmed in the blockchain. It isn't "harder to do", and even if you control the miner you can't "theoretically make it go through". Only one transaction spending the unspent outputs can permanently exist in the blockchain. All other transactions that attempt to spend those outputs will be rejected by all peers.
It makes sure you follow the correct chain I think?
MultiBit works fine without storing the entire chain.
I read it helps to prevent doublespend, but wasn't that already fixed in an update to Bitcoin-Qt?
Correct. In order for Bitcoin to work, there needs to be enough people running full nodes such as Bitcoin-Qt (containing the entire blockchain). The blockchain makes double-spending impossible and provides a way for a decentralized system to reach consensus on the order of transactions. As long as there are enough nodes with the full blockchain to keep everything secure, then other users can run wallets that use SPV and rely on those peers running full nodes for the information they need.
And what're the advantages of downloading the whole block? I read it helps to prevent doublespend, but wasn't that already fixed in an update to Bitcoin-Qt?
There are a few, difficult to perform, theoretical attacks that someone can perform against you if you are only running an SPV wallet.
https://en.bitcoin.it/wiki/Thin_Client_Security