ji,
My opinion is like during regular puja at home it is not necessary to mention these details. in case of special occassions most of the cases vadhyar will be present and he will guide. below are few points in this regard.
1. The order is Gothra star then rashi and then name .
2. while mentioning name give the name what is given during naming ceremony. The sharma name for boys and the given name for girls . add the term "sharma" after boys names.(each varna has their respective name like sharma)
3. In temples start with the eldest living person of the family then end with the younger. No need to mention the details of born babies till 3rd month . in few families it is not till the 1st birthday.
4. In houses on special occasions it will start with the person who doing the sankalpa and follows the order of spouse , children and then parents
5. If you want to add in laws family or the family of any other gothram , first finish all the members of one gothra and then move to the other.
Thanks,
Arun M