I'm glad you figured it out.
Unfortunately, I cannot help you with the customizations. AS is using Bootstrap 4.1 and FontAwesome 5, and if your website is using a different version of bootstrap or font awesome, then you will need to either modify AS's HTML or to make your website use the same version of bootstrap and font awesome.
Another option is to have the completely custom login form which will post the data via AJAX to the same endpoint that the original AS login form does. Also, instead of embedding the login form to your website, you can always redirect to the login page where users can log into the app.