Showing categories in Channel Entry URLs
There's a lot here but I'd start by create 2 different channel sections: Blog and News. 

From the front end, that makes it easy to see which entries are associated with what sections and easy to post content into each one. It also makes it easy to filter in your templates what section you're after.

To get your entry URLs setup, it's a little tricker but you can pull it off with some Craft/Twig magic. Setup a new field, make the handle blogCategory, make it a Category relational field, lock it down to 1 category that pulls from your Blog Category Group, and assign it as a field to your Blog section. 



When you're creating the channel, set the URL format in your Channel to this:

blog/{blogCategory.first().slug}/{slug}




That little bit of code basically tells Craft to use the category slug of the blogCategorycustom field we just created in part of the url and the slug of the current entry to make the url. Craft treats categories, assets, entries, etc. and other relational fields as an "array" even if you only have one selected so we need to use .first() to use the first one Craft sees.

Now when you go create your blog entry, you're able to set a blog category and Craft will also use it as part of the URL.

BTW: There's no problem routing the Blog/News index pages to a page in a structure but you have to do the heavy lifting that Craft usually automatically routes for you.

In your _layout.html template, you could do something like:

{%if craft.request.firstSegment() is "news" %}
    {#load up news template #}
    {% include '_news/index.html' %}
{%elseif craft.request.firstSegment() is "blog" %}
    {#load up news template #}
    {% include '_blog/index.html' %}
{%endif%}


There's more to it than that, of course but hopefully that might get you started. It takes a little while to get your head around Twig and how powerful it is to use in Craft.

You'll want to also setup URLs for your Categories to have Craft load the appropriate template for each. Understanding the basics of how Craft routes requests is helpful as well.