Counterparty is based upon PoB - Proof of Burn concept. You can check the total XCP created during the burning period here:
http://blockscan.com/burn.aspxWhat I'm wondering is if it's a mined coin with transactions handled by BTC block chain
or do the transactions get verified via proof of stake or some other way?
There is no mining. All the XCP coins that will ever exist had been created by burning the BTCs in January. All the transactions will be stored in BTC blockchain. There is no proof of stake mining too. The only way you gain by investing in XCP is if the price of XCP goes up.
XCP is deflationary. 0.5 XCP will be destroyed for each user created assets.