How is a group element converted into a key?
I don't know the protocol, so I'm not sure. 
It is a standard way of converting group elements (e.g. the shared value $g^{ab}$ you get after Diffie-Hellman) to keys. You don't need to worry about the specifics of the group when choosing a hash function, just use a cryptographic hash like SHA256.
It's hard to know what the difference between those two is without more context. You are certainly using the hash function to derive a key, but not all hash functions are also key derivation functions. Usually you talk about 'extractors' in the context of extracting entropy from a value, so in a sense the hash function is also an 'extractor' in that the hash is (probably) shorter than the value being hashed but contains (roughly) the same amount of entropy.