I am trying to add a newsletter subscribe form that looks like this:
Is it possible to use custom CSS to add the field?
You need to add your check box in the form first, which supposes that you use a field in the database.
Once this is done, you can use some CSS and even some JS to move the elements around.
No, you can also use the JS on the embedded form. You need to add the specific code to the embedded form JS itself, as described in the forms API documentation: