The groups you are considering are a special class of Shephard groups, which are obtained from Artin groups by adding relators $\sigma_i^{k_i}=1$, $0\le k_i<\infty$ for every Artin generator. Biautomaticity of some of all these groups is conjectured in http://arxiv.org/abs/0901.0094 and this conjecture is verified in some cases. These results apply to "most" but not all groups that you are interested in.

The groups you are considering are a special class of Shephard groups, which are obtained from Artin groups by adding relators $\sigma_i^{k_i}=1$, $0\le k_i<\infty$ for every Artin generator. Biautomaticity of some of all these groups is conjectured in http://arxiv.org/abs/0901.0094 and this conjecture is verified in some cases. Unfortunately for you, the results in this paper require that the Artin graph contains no triangles with label 2, which will exclude quotients of classical braid groups. Nevertheless, you may want to read the paper to see if the methods could be useful in the context of your question.

The groups you are considering are a special class of Shephard groups, which are obtained from Artin groups by adding relators $\sigma_i^{k_i}=1$, $0\le k_i<\infty$ for every Artin generator. Biautomaticity of some of all these groups is conjectured in http://arxiv.org/abs/0901.0094 and this conjecture is verified in some cases. You may want to read this paper closely to see if it says anything interesting about the case when Artin's group is the classical braid group.

The groups you are considering are a special class of Shephard groups, which are obtained from Artin groups by adding relators $\sigma_i^{k_i}=1$, $0\le k_i<\infty$ for every Artin generator. Biautomaticity of some of all these groups is conjectured in http://arxiv.org/abs/0901.0094 and this conjecture is verified in some cases. These results apply to "most" but not all groups that you are interested in.