I think, personally, this UI fits in really nicely with Review Board's pre-existing metaphors. Great job!
A few notes:

We probably still want the filename in the center of this, rather than pushed to the right.
The labels should be vertical-center aligned
Either both should end with : or not.
We probably want a little extra space between "Custom" and the image on its left - maybe the same amount of distance between the background color selectors.

You're working with some older code that predates our modern CSS naming conventions, but it'd still be good to make use of them for all new class names. See our documentation on that.

The text-align should predecede the nested table rule, because it's relevant to .review-ui-header.
You'd also want a blank line between that and nested rules.

Can you put this in alphabetical order? While not always appropriate for all CSS rules, it's a good point to start. The exception to it is that we group related properties (width/height, position/top/right/bottom/left, etc.) together.
This applies to other rules below.

This is too generic a name, so it'll be especially important to namespace this, as per our docs.

Blank line between these.

Can you do testing on Firefox and Safari for these?

Can you reorder these to be in alphabetical order?

The blank line here should stay.

Some important comments here:

Keep line lengths in mind. Lines cannot go beyond 79 characters.
These sorts of CSS changes are best done in CSS, not in JavaScript.
Always use const or let instead of var.

Blank line between these.

Space after the if, before the (.

Can you go through the file and make sure there's no extra whitespace (identified by these red blocks)?

This would be best done as a _.template somewhere, using _.dedent (see other parts of the codebase for examples).

Any text must be localized and provided in. This is easier done with templates. You can localize a string using:

_`Background`

Always use double quotes for attribute values, and never self-close tags.

</> isn't valid.

You can use chaining to reduce the code here:

$('<div class="....">')
    .append($backgroundMenu)
    .append($resolutionMenu)
    .appendTo($header);

You can set multiple model attributes at once:

this.model.set({
    checkeredBackground: false,
    background: color,
});


This helps with ensuring that signal handlers can get the new set of attributes at the same time, rather than triggering a bunch of different signals and leaving it up to the code to sort it out.
While not an issue necessarily right here, it's a pattern that's good to adopt for Backbone.

This can be one statement.
Also, code should always use single quotes instead of double quotes, unless the text in the string requires a single quote.
These apply to code below.

Missing a period.
This must also be one line (and can't go beyond the line length).

This should probably be driven by the model.
The model should listen for a change to checkeredBackground. If set, it should set the background to a value.
It should probably also go the other way. If explicitly setting a background, unset checkeredBackground. That would simplify code above.

Please wrap your description and testing done sections to be 80 characters wide.

Let's use #rb-ns-ui.colors[@grey-40] instead of #808080

This should also use one of the #rb-ns-ui.colors definitions (see reviewboard/static/rb/css/ui/colors.less) instead of "grey"

Please use single quotes instead of double.

Please use single quotes instead of double.

This should probably be called backgroundMenuTemplate. The $ prefix is for jQuery objects.

We could skip defining this as a variable and just do:

$('<div class="...">')
    .append(...)

Let's wrap this like so:

.append(backgroundMenuTemplate({
    backgroundLabel: backgroundLabel,
    customLabel: customLabel,
})
.append($resolutionMenu)
...


Note that we also don't need quotes around the keys that we're passing in there.

Please add a trailing comma at the end of this.

Please also go through and mark "Fixed" on any old issues which have been fixed up.

Hi Matthew! It is an awesome feature that you implement!  I was wondering why in the styling sheets, sometimes you are using the tag identifier started with the & sign, such as &.-black and &.-white, instead of the typical .-white? Also, for the checkered background selection, why do we need to specify the background-position and background-size, while the other selections don't?

I think we should have the captions be last.
We want to think of the UI in logical components. We have here, in order:

Controls for manipulating the view
Labels that describe the content below
Controls for manipulating the view
Content

It'd be much nicer as:

Controls for manipulating the view
Labels that describe the content below
Content

I only see one place each where we use these. We don't need to define new variables for them. Let's put them where they're needed.

A more standardized name (in our modern standard) would be -has-checkered-background.

Do we need !important?

Blank line between these.

We usually make display the first, or one of the first, properties in a rule. Can you move it up?

Where possible, let's use em units. These are going to scale better than pixel units, making them more accessible for users.
Usually we have units in increments of 0.5em, depending on what we're displaying (e.g., 1em, 2em, 1.5em, etc.). Not a hard-and-fast rule, but that does help keep a visual consistency.
Same throughout the change.

No blank line here.

These are looking pretty long. Not uncommon for this to happen.
Try:

events: {
    ...
    'click .rb-o-image-background-blah-blah-blah':
        '_onBlahChanged',
    ...
}

When you're setting a single attribute, pass each in as a parameter:

this.model.set('checkeredBackground', false);

this.$el is the .image-review-ui component, so rather than scanning the entire DOM for it each time, you can just use this.$el.

This can now be:

this.$el.toggleClass('-has-checkered-background',
                     isCheckeredBackground);

All this should be 1-space indentation.

Only one space indentation.
This should also be <%- backgroundLabel %>, in case the text gets localized to something that isn't HTML-safe (<%- will escape text).

The /> must be a >.

for must align with class.

This must also be <%-.

This is indented 1 space too few.

Rather than defining the variables above, you can pass in the strings here directly.

Missing comma after this last line in the list.

This could be one statement.

This could be one statement.

Doc summaries must be a single line.
You can go into more detail in the description.
I'd just say "changed to a checkered pattern."