Custom form from scratch in template I have a paragraph that I fully themed into a contact form with the necessary  tags etc... Now my question is: How do I add the actual functionality to this? Because right now it's just plain html in a template without any logic handling the form submit.
You can use Drupal's default contact form and customize the fields as per your need which will have all the required logic for form handling. Then you can check and apply the theme as per your need to it.