Yes it does, the project runs a reference node.
The PoW comes from Bitcoin, but anyone can run a Counterparty node. I don't know if there's a name for it, but maybe it could be described as a community consensus approach. All Counterparty nodes can validate transactions embedded in the blockchain.
The purpose of burning BTC was to make it a non-scam token. Those who burned BTC committed to the project. Another way to put it is it's a project without "investors". Proof of Burn was rare then and in the flood of ICO scams, it's never applied these days.
Where is the link to run your own counter party node? – Patoshi パトシ – 2017-10-22T19:22:53.617
https://github.com/CounterpartyXCP/Documentation/blob/master/Installation/federated_node.md - this installs everything in one go, inside of Ubuntu 16.04 LTS VM. Be sure you have 150GB free (because this also installs Bitcoin Core with addindex and txindex). – whatever – 2017-10-27T22:19:49.907