To handle the single domain, you’ll want to obtain hosting that will run both your currently Azure-hosted site and WordPress. Usually Linux is the easiest way to go, though WP can run on IIS.
You basically need to migrate the existing Azure-hosted site. Then, you can install WordPress in either a subdomain or a subfolder. If you want the two sites to appear to be one, a subfolder is your best bet.
Note, WordPress.com is a fully hosted solution that’s very limited (and not actually supported on this site). What you’ll need to do is set up self-hosted WordPress from WordPress.org and then migrate over your content. You can then build a custom theme so that your header and footer on the “other” site and the WordPress site are seamless.
As far as embedding content from the “other” site, it really depends on what type of content you’re trying to embed and what APIs are available on the “other” end. When you set up the custom theme, you can also build in functionality to pull in external data through APIs or RSS feeds. If you’re placing specific content in specific places, you can accomplish this with PHP and JS in the theme. If instead you want to embed content in various places, you may want to look into building WP Blocks, which use React, and will allow you to add content in any post/page/CPT in any part of the main content area.