I see you've made changes to the visual style, so can you add a screenshot or screenshots showing these changes?

Class names should be kebab-case e.g. ship-it-checkbox-area.

This is needlessly specific. .ship-it-checkbox would be fine.
The more specific a selector is, the more specific a selector that would override it must be, which can lead to unmaintainable CSS.

We can restructure this as:

.ship-it-checkbox {
  display: none;

  &:not(:checked) ~ .rb-icon-admin-enabled {
    // ...
  }

  &:checked ~ .rb-icon-admin-enabled {
    // ...
  }

  &:checked ~ .rb-icon-admin-disabled {
    // ...
  }
}

Why this change? Isn't this the label for the body top editor?
If this is correct, gettext('') will always be ''.

Undo this.

Why this change?

I don't know that this is the right text for this.

Can this float: right style be added to the stylesheet?

In this section you have about four <label> elements for one input (for="id_shipit). While it is technically possible to have multiple labels for one form element, there are some accessibility concerns with this as many assistive technologies, like screenreaders, don't know how to handle them.
I'm not sure what the best approach is in this particular case so you might need to do some playing so you only have one for= for the one input. (There's an example here that could be helpful: https://developer.mozilla.org/en-US/docs/Learn/HTML/Forms/How_to_structure_an_HTML_form#Multiple_labels)

Your description mentions this a WIP commit

Regarding testing done, I think you could run the unit tests for the review dialog to make sure everything still works fine. There were also variable name changes in the test files so it will be good to see that the tests still run properly.

If you feel like your changes do not need a unit test, you could just state what you manually tested and how it works fine.

I noticed in the screenshot the edit icon (pencil) wasn't to the far right as shown in the design document. You can find change the position in the inlineEditorView.es6.js, if you search for class="editicon" you should find the element you need to get the desired effect

You should add a semicolon here

You should add a semicolon here

You can remove the bit about the tests on master failing from your testing done.