Is there a way to register watched addresses in the standard client?
Not yet. There is a pull request implementing bloom filters that should make that easy to implement.
Or, could 'listunspent' be extended to take any non-wallet address as an optional parameter?
No. The reference implementation doesn't keep a master map of all addresses to unspent transaction outputs, and adding such an index for the small number of services that need to look up arbitrary addresses doesn't make sense.
I'd suggest you -blocknotify and the getblock() RPC call to maintain your own index of address --> unspent txout.
Even getblock is used, it only returns the txid, and gettransaction does not show the tx if is a non-wallet id. What should I do?