Demand function of a family
I think any (and none) of your answers is correct. The market demand simply depends on how the family makes its purchasing decisions.

For example, suppose the matriarch $i$ decides on every purchase of the family and acts selfishly. Then obviously the utility function of the family is simply $u_i$.

Compare this with the case where the utility functions are cardinally comparable and the household is utilitarian. In this case, the family utility function is the sum of all utility functions.

More generally speaking, if purchase decisions are made via some game, then you have to solve the game first in order to obtain the demand functions. But whether a utility representation exists for the family utility and what it would look like crucially depends on the game structure.

One special case which has been discussed in the literature is the case where the household income is somehow distributed among family members and each family member spends separately. There then exists a class of utility functions for which the family expenditure will be independent of the distribution of income among the family members. For details, see https://en.wikipedia.org/wiki/Gorman_polar_form