The problem with this approach is that the deposit can be spent to another address in the same block as the double spend.
No, that's no problem: usually miners will not do this as they can earn more by using Tx3 (or else, the guy who double spends still pays the miner a lot of fees).
The problem TalkingAntColony pointed out is that of specialized "double spend miners". These build reputation and make money by helping the double spender. In the long run they can make more money by helping the double spender and not use TX3 but by mining the double spend and moving the deposit.
Sometimes (~1%) the double spend miner's blocks get orphaned, though, which means the TXs from the block become public and other miners will mine TX3.
Nodes could delay relaying of blocks including potential double spends (txs replacing txs the node knows about).
This would give miners an incentive to use the first / most popular tx with a certain prev output.
Do nodes currently not transmit tx's which are double spends, i.e., only keep the first one?
Yes. Invalid TXs will be silently ignored by the standard client. There were discussions about broadcasting double spend warnings along with the two conflicting TXs but there is only little benefit for a careful merchant.