Where can I query the UTXO database?
The standard bitcoind core daemon has no methods to query the UTXO set for reasons I don't know: it should be trivial to give access to it even on a pruned node since it needs to have the full UTXO set at any time to validate a new block.

Anyway a pull request seems in progress to add this capability:
https://github.com/bitcoin/bitcoin/pull/8952

There are anyway third party apis to query a single address, for example https://www.blockcypher.com/dev/ or https://chain.so/api