"People" is plural so you should use "their".
"Before all family washes his OR their hands for meal, they spread the table." This sentence should be "Before all of the family wash their hands for the meal, they set the table."
The word "all" is plural so use "their".
In the last case, group is singular, but it is understood that you are talking about the members (plural) of the group, so you should use "their autographs"