Are there counterparty nodes that one can run to support the counterparty network? Does counterparty have its own set of nodes to keep its service up? 
Is it pow or pos system? Or is this just a centralized service? 
What was the purpose of it burning bitcoins to generate their own counter party tokens when its somewhat centralized?
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.