From my understanding the transaction will hang around for some time, but most miners ignore malicious transactions like this so it is essentially discarded.
I believe you could resubmit the transaction after funding, I've seen people modify stalled transactions to include omitted transaction fees.
Are you "missing" BTC because you tried to manually push an invalid transaction? I think there is a simple command that should make your wallet recheck all your addresses, which would make the coins "appear" again.
No. I've just seen unconfirmed transactions on the blockchain explorer (@ blockchain.info) which have hung around as unconfirmed for days because the address appeared to have insufficient funds at least that is why I thought it was marked as unconfirmed.
In my opinion, the blockchain explorer should note the transaction as invalid or invalid with reason listed; maybe it works this way, but I suspect that it is just discarded. Are invalid transactions propagated through the whole p2p network even if invalid?