Formatting email sent with Contact form
Putting together an answer based on the comments/discussion there so this can be accepted/closed.


General instructions on how to theme a mail can be found in this existing answer: How can I theme the outgoing contact mail?
You might want to watch https://www.drupal.org/node/2680731, once resolved, you can rely on that instead of your own custom code to send the mail as HTML.
To use a custom template for the inner parts of the mail, you can set a custom template in hook_contact_message_view(). Or use field templates for specific fields. 
To check for a specific contact form, you can use the bundle() method of the contact message entity, which you can find in $message['params]['contact_message'] in hook_mail_alter(). Make sure you combine this with a check that it is there and an entity object, otherwise you could have fatal errors for other mails.