Sextic twist of BN pairing parameters vs security
A current conservative estimate of the security level of a BN curve with an extension field size of 3072 bits, taking into account the Kim-Barbulescu attacks, is 2110. See the paper http://eprint.iacr.org/2016/1102.pdf and the discussion of it at https://github.com/zcash/zcash/issues/714#issuecomment-263007229 . An earlier discussion of these attacks is https://ellipticnews.wordpress.com/2016/05/02/kim-barbulescu-variant-of-the-number-field-sieve-to-compute-discrete-logarithms-in-finite-fields/ .

On the specific question about the sextic twist, G_2 is still an elliptic curve group, not the multiplicative group of a finite field. So index calculus attacks, including variants of TNFS based on Kim and Barbulescu's work, do not apply to it.