Are there counterparty nodes that one can run to support the counterparty network?
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.