Or is this what payment processors do already and we are talking about something quicker here?
Roughly spoken, yes they do this already. They scan for any incoming payments, and when they see one, they try to detect any conflicting payments (double spend attempts). If that doesn't occur for a few seconds, they assume it's safe to accept the transaction.
There's a lot of theoretical debate around this (especially involving mining pools running customized software, such as Eligius and BitUndo) but in every day practice this turns out to work extremely well and very secure.
No, no, NO! A common misconception.
1. Other coins do NOT have faster transaction times. Bitcoin's P2P network is by far the biggest, and (unlike centralized models and networks) the more nodes and connections, the faster it distributes data. Bitcoin transactions are faster than any other coin.
2. Other coins also do NOT have faster confirmation times when measuring certainty against waiting time (i.e. the only thing that really matters). People confuse certainty (probability that a transactions remains final) with the number of blocks that include the transaction, without taking into account how much processing is actually required to find such one block. Yes, the more blocks a transaction, the higher its certainty, but Bitcoin provides MUCH more certainty per block than other coins.
For example, Litecoin has 4x faster blocks (every 2.5 minutes instead of 10). This means the average calculation time to find a block is 4x less. This also means it's 4x easier to 'undo' a block (mine an alternative block to fork the blockchain). This means 1 Bitcoin block would offer the same amount of certainty as 4 Litecoins blocks! And that still only holds when Bitcoin and Litecoin networks would have equal hashing power. But that's not the case: Bitcoin's total hashing power is MUCH more than Litecoin's or even all other coins combined. So in fact, one Bitcoin block provides even MUCH more security than 4 Litecoin blocks!
TL;DR = Bitcoin wins.