Where can I query the UTXO database? Currently there are 35 million unspent transaction outputs. Is there a site somewhere where I can query this data to see, for instance, how many UTXO's exist with a value less than 20,000 satoshis? Or the total unique scriptPubKey values found in a  UTXOs?
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