Is it permissible to fast six days of Shawwal and another consecutive day to fulfill a vow of fasting seven consecutive days? And is there a comprehensive jurisprudential rule for combining the fasting of a vow with supererogatory and obligatory fasts?

It is not recommended to make a vow, but whoever vows an act of obedience must fulfill it.

The most preferable and best approach for fasting a vow and the six days of Shawwal is to fast the vow alone and to fast the six days of Shawwal with the intention of the vow. This way, one obtains the reward for fasting the vow and the six days together, but not the reward for someone who fasted the six days of Shawwal independently. It is not possible to combine fasting a vow with other obligatory fasts or supererogatory fasts intended for themselves.