to the second question.Once the child/ren moves out, it will get reassessed.

Does anyone know which states are similar to CA Prop 13...in terms of fixed increases in property assessment value for property tax and able to pass down the assessment value to children?

Prop 19 is pretty limiting now with allowing children to retain a parent's assessed value for real property. 

When she passes, the house will be left to 4 adult children, but with the passing of Prop 19 on April 1, 2021, 1 of the 4 children will have to claim it as their primary residence to keep my mother's tax basis.

@Rob K. response seems very good, but I want to add that one of the two proposed prop 13 amendment initiatives (prop 19) eliminates the Prop 13 tax bases transferring to children and grandchildren if the unit is not the primary residence (I.e. rentals and vacation/second homes).  

@Ryan Espinosa Newly passed Proposition 19 will have potentially severe financial consequences for children inheriting property from their parents.

They had no vehicle; the driveway had not been plowed since they moved in but somehow, they could get fuel.Anyway, they sat at the curb with three young children for over 2 hours in 27-degree weather before they finally got a ride to get a U-Haul.

In Norman for instance, you see a lot of parents buying properties for their children to live in during school, but they buy them at owner-occupant prices and it drives up the comps for everything.

Units will be so assigned that:Children age four and under will not be allotted an additional bedroom with any other child or a parent, regardless of age or sex;Two children between the ages of five and seventeen of the same sex will not be allotted an additional bedroom;Two children between the ages of five and seventeen of the opposite sex will be allotted an additional bedroom, although they may share a bedroom at the request of the household.Adults (over age eighteen) of the same sex will not be allotted an additional bedroom;Adults (over age eighteen) of opposite sexes who are spouses or co-heads will not be allotted an additional bedroom;Adults (over age eighteen) of opposite sexes who are not spouses or co-heads of household will be allotted an additional bedroom although they may share a bedroom at the request of the household.A single head of household parent will be allotted an additional bedroom for with his/her child over age four, although they may share a bedroom at the request of the household.Exceptions to the largest permissible apartment size may be made in case of an approved reasonable accommodations for a person with disabilities.An unborn child will be counted as a person in determining apartment size.

For instance, if you want to rent to traveling medical professionals make sure to buy near Texas Children’s Hospital or another Level 1 trauma center.