Integration

@khorkch - Welcome to Backdrop CMS community. Here is my take on the answer to your questions. Hopefully, others will help me out.

Is this CMS able to integrate into our existing templates?

It's possible to create your own templates for Backdrop CMS or edit the existing templates. I don't believe that there is any automated way to pull in your existing templates, but you certainly could recreate them within the CMS. This would probably require at least a little coding or html skills. 

Is my client able to edit website on their side?

Backdrop comes with very flexible "roles" and very customizable "permissions." Which means that you can create custom roles for your clients to edit their own site and you can be very fine grained about which permissions you give them. So you decide which types of content they can add or edit and which they can't.

https://backdropcms.org/user-guide/users-roles-and-permissions

Is the CMS able to integrate into our .net platform?

The short answer is that I believe that it's possible to integrate Backdrop CMS with anything if you have the tech skills and budget. But, that's not a very helpful answer. 

I'm not sure how easy it would be or what you would need. Can you say anything more about what kind of integration you would be looking for?

Is the CMS able to integrate into our .net platform?

Here are Backdrop's system requirements: https://backdropcms.org/requirements?destination=admin/dashboard I don't know if it's possible to run PHP and MySQL on a .net system, but if so then the answer is "yes".