How can I get all child categories - of a child category based on url segment?
There are a couple of things I would do differently here. The first is to check the Categories in this group have their own URLs box in the Category Group settings. Set the url to {slug}, and you will have category defined in the template without needing:


  {% set categoryGroup = craft.categories.group('Sections')
  .level(1)
  .slug(craft.request.getSegment(1))
  .first() %}


The second problem is that once you have your category, what you want are the children, not the ancestors:

{% set theChildren = category.children %}

{% if theChildren|length %}
  {% for cat in theChildren %}
    {{ cat.title }}
  {% endfor %}
{% else %}
  NOTHING
{% endif %}