Children taken

Guardianship - temporary or permanent

if she doesn't abandon them or CPS takes custody by court order she can place them anywhere she pleases. Fostering only occurs when abandoned or taken away by court order.

If the children are going to attend school, proof of guardianship and address will be needed. So a child can go live with grandma and attend school in grandma’s school district.

The court will grant guardianship and that can be transferred back to the parent when they have met cps requirements.

If the police actually contacted CPS, and CPS placed them with the first family members, then they are probably in foster care, but CPS did a child specific license for the family members. If CPS is involved and they do the licensing process with the next family members, the children are still in foster care. They do not need to be in a general licensed home to be considered "foster children".

It depends. Are you related to the children that you are taking guardianship of? Are the parents rights being terminated?
I say it is always better to get permanent guardianship. If you feel later on that the parents are ok to have the children reunified with them, and the CPS case is closed, then you as the Guardian, and the parent can petition the family courts on your own to terminate the guardianship and have the children returned to the parents. If you never feel it is ok for the children to return to the parents, then you don't have to do this. If the parents rights were terminated, then you wouldn't have to worry about this anyway, because the children wouldn't be reunified, even if later you thought it would be a good idea.