If a parent dies and the surviving parent assumes all property while having two children but one is still a minor (13). Then the surviving parent re-marries and adds the new spouse's name to property. Later on in years the property is sold. What part of the deceased parents' property still belongs to the children? If the last parent dies and everything has been sold or transferred into the new spouses' name, what rights and or inheritance do the children have?

Whatever portion of the original property was left to the children in the original will.

Most likely though, the surviving spouse took all property rights and the kids are due what she wills to them.

There is not way to answer this question without knowing how the surviving spouse gained control of the property.

If it was through a will or joint tenancy, then once the property was transferred, its up to the current owner how to distribute it. If the property has now been transferred to the new spouse the children are probably out of luck.