Use pure PHP and HTML instead of Templates
You're really fighting against the way Craft CMS does things. 

I'd start by having a good read through the Craft Docs first, there really isn't a shortcut. The docs are really great though, and contain everything you're likely to need.

Then, I'd set up a copy of Craft in isolation of your site/framework and just get a feel for how you do things the "Craft way". It is different to just doing HTML &amp; PHP, but not so different that's it going to take you too long to figure it out. 

Craft actually uses the Twig tempting language, which is really easy to get going with.

In answer to your specific question...

Does craft allow me to make calls to get the stored data and then I manually handle the inserting of this data?

Yes it does. To output your data, you'd likely use sections/entries in your Twig template. So take a look at the example at the top of this page to see how you'd go about it. You'd input your data via the Control Panel, so have a read here for that.