How big is the UTXO database?
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.