Both Tag Manager and gtag.js perform similar functions:

• Tag Manager is a tag management system that allows you to quickly and easily update tags on your website or mobile app from a web interface.
• gtag.js is the JavaScript framework that is used to add Google tags directly to web pages.

Both solutions are built on the same core Google infrastructure, and are interoperable. You are free to choose the solution that best suits your needs.

Consider Tag Manager if you need:

• The ability to deploy and modify both Google and 3rd party tags
• The ability to deploy and modify tags for both web and mobile apps
• The ability to deploy and modify tags on the fly from a web interface
• Collaboration and versioning capabilities

Consider gtag.js if you are unable to use Google Tag Manager, or the following capabilities are important to you:

• The ability to install tags directly on your web page, without the ramp-up time of setting up a tag management system
• The ability to work with your tags directly in JavaScript without being required to work with a separate interface

https://support.google.com/tagmanager/answer/7582054?hl=en

No. But you can create an Adwords conversion tag within GTM (there is a template for this, the part before the slash is the conversion id, the part after the slash the conversion label). If you want you can configure the tag via the dataLayer, but you'd still have to create a tag within GTM and apply a trigger to make this work.