AFAIK, that's effectively what mining is. From my understanding, transactions exist in a limbo state (where they're relayed around the bitcoin network) until they're included in a mined block and become part of the blockchain. A "mined block" is basically just a list of transactions (including a coinbase transaction, which is the new bitcoins), the merkel root of the previous blocks, and some other data such that the hash of the entire thing is suitable, as determined by the network difficulty. Thus, mining and confirming transactions are effectively one and the same.
Sorry for the poor (and possibly incorrect) explanation, I'm still wrapping my head around it too.
1) The hashing difficulty automatically adjusts to match the network throughput, so that new blocks are found once every 10 minutes. New blocks introduce new Bitcoins, so this controls the total supply.
2) New blocks extend the last agreed-upon block. If an attacker forged a block, they would have to do so faster than the rest of the network combined to control the entire block chain (clients agree the longest block chain is legitimate). While you could do this without the same computational expense, it would rely much more on Internet connectivity.
An older AMD GPU can calculate ~250 million hashes per second. With no difficulty, a transaction could be verified on the network as fast as miners could broadcast their agreement. This would mean attackers could more easily control the block chain with DDoS and other network-related tricks. Even now, the largest mining pools are sometimes targeted to slow down transaction processing.
It's kind of like captchas - a seemingly pointless pain in the ass on purpose to weed out the spam.