Is Augur's reputation token a cryptocurrency in traditional sense, or not? Is it based on Bitcoin's source code?
REP is a token, not a crypto-currency (though the lines between the two are a bit blurred).
Augur is not based on Bitcoin source code, it has its own code base and is built on top of Ethereum.
REP is an ERC20 token contract deployed to the Ethereum blockchain.
ERC stands for Ethereum Requests for Comments and the number 20 is the first successful attempt to standardize the token interface for Ethereum-based tokens.
To answer your initial questions, Yes, REP is a crypto-currency and can be used as such, but No, it's not based on the Bitcoin source code, and not even related to Bitcoin at all.