How do I know if a given curve requires a FpCurve F2mCurve or ECCCurve?
The EC objects implement general elliptic curve properties, whereas the other 2 operate on specific fields. Since secp256k1 is a prime field curve, you want the Fp objects. For binary field curves, you would use the F2m objects.

Note that NIST "K" curves are binary field curves, wheras secp256k1 is a "K" curve but over a prime field. Koblitz curves allow efficient mathematical operations.