How big is the UTXO database? Is there a query that will return the size of the UTXO database? Specifically, the total number of unspent outputs.
Bitcoin Core provides the gettxoutsetinfo RPC that has exactly the information you want.  Note, it takes that call up to a minute or two to run (and maybe longer as the UTXO set keeps growing).

Here's what my node says right now:

{
  "height": 442313,
  "transactions": 15237948,
  "txouts": 43194647,
  "bytes_serialized": 1635888675,
  "total_amount": 16028761.10201779
}


Current and historical data may be found on Statoshi.info's UTXO page.