Is it permissible for a divorced woman to go for Umrah accompanied by her children, and can she perform Hajj alone, knowing that she has no brothers and her father is elderly?

If a boy has not reached puberty, he cannot be a mahram (definition) for a woman on her journey, as a mahram is required to be an adult. However, if a woman has not yet performed Hajj al-Islam or Umrah (definition), the more correct view is that it is permissible for her to travel for Umrah and the obligatory Hajj with trustworthy company.