Posts appearing under categories

My site has five major categories which are referenced on the main menu, each with it’s own page. Let’s call the categories A, B, C, D and E (the actual terms are SEO-friendly).

When I create blog posts they are automatically created under /blogs. I’d like to be able to set a “parent page” for posts, in the same way can for pages. For example, I’d like posts to show up with URLs like:

/A/post-about-something-to-do-with-A
/A/post-article-about-something-to-do-with-A
/B/post-about-B

And so on. I don’t really need a single parent (to replace /blog) because I have a number of categories. But I’d like to be able to define the parent of the post when I create it.

By the way, when I press the Edit button to edit the URL for a particular post, I can edit the name of the post but can’t edit the /blog part.

Is this possible?