Is it obligatory to make up the remaining days of Ramadan on behalf of a deceased father, and is it permissible to combine the intention of fasting the six days of Shawwal with making up those days?

We understand from the question that the days your father missed do not need to be made up, as he passed away before them, and the deceased are not held accountable. It is not permissible to combine the intention of making up Ramadan with the intention of fasting the six days of Shawwal, because they are two distinct acts of worship intended for their own sake.