Magento – Why magento 1.9 not showing newly added categories

This is actually a feature. If you want to avoid this you should set the flag 'Is anchor' to 'No' on the parent category. In your example 'Clothing'. This way, in the parent category you will see only products directly assigned to it.
After you do the changes you should re-index everything.

This sounds like you havn't configured your categories / products correctly, or else it should work in the default theme. I'd go a step back, configure your categories & products so they show in the default theme, then switch themes.

Part2:

Normally a homepage will have different widgets showing latest products, imagesliders for promotions and anything else you want there to make the initial customer impression.

You can either edit your existing front page or create a new one, or use a category as the store front.

On the homepage you would have to configure or edit your cms front page via cms -> pages and insert a widget to show products of a certain category

eg:

{{block type="catalog/product_list" category_id="8" template="catalog/product/list.phtml"}}

Or if you wish to make a category your front page, get the category ID of the one you wish top make your store front and go to "System > Configuration > Web > Default Pages" and enter the following for the "Default web url" option: catalog/category/view/id/10 (where 10 is the id of the category).