Is there any tool to index ALL bitcoind UTXO? I've got a full bitcoind node with txindex=1.

listunspent command is useful only if node is in possession of a private key for address.
So, if I want to check UTXO for random address, it will not work until all of that UTXO is indexed properly.

So, are there any tools to do that simultaneously with bitcoind?

Of course there is an option to import that private key to bitcoind. But this question is larger - to get unspent for any address.
This is something that bitcoind doesn't do, but there are block explorer tools that work on top of it that will give you address-specific utxos:


https://github.com/bitpay/insight
https://github.com/coinbase/toshi
https://github.com/bitcoin-abe/bitcoin-abe