The emails that go out are very plain looking. We want to integrate our brand with the messages that go out. When I use the HTML view when editing a template, and then save, most of my HTML get stripped out. 

Is there a single template I edit (like through a hook/filter) that would allow me to the text that is in the default notification with a design (something basic like a messages WooCommerce sends). If there is no template, is there a way to stop my HTML from being stripped?

Standard HTML tags:

div, h1, p, a, img, table, tr, td, strong

Also, if there's a better way to do these emails than to add HTML to every single message. Can with hook into a filter or action that will output a wrapper, so that the content in the email panel will be inserted into the template?

I find that shocking. Tables have long been the most reliable structure for email newsletters and messages, since there are so many email clients that use old code. I have done as you asked - I replaced all of the related table tags with div tags. When testing this, all of the divs were removed, just as it ddid when I first submitted the ticket. If you click Save, then refresh the page, then click on the notification, you will see all the <div> are converted to <p> tags and all the other tags are removed except for <p><br><a><strong>. This seems very incomplete or broken.

Also, you did not touch on my part of the question: "Also, if there's a better way to do these emails than to add HTML to every single message. Can we hook into a filter or action that will output a wrapper, so that the content in the email panel will be inserted into the template? (like add a header/footer to all notifications)"