is there a way to set a background color for an article with jce editor?

You could probably do this by wrapping all the content in a div tag with a special class attribute that styles the div (best done in the Code editor), eg:

<div class="content-bg-1">
Your content...
</div>

then add the class to your template stylesheet, eg:

.content-bg-1 {
    background-color: #eee;
}

but how effective this is will depend on how your template lays out the article content.

Or is it possible to set a background color or an image as a background in a block?

Assuming you mean a column created using the Columns tool, again the best way is to use a css style in your template stylesheet, eg:

.content-bg-1 {
    background-color: #eee;
}

and then to select and apply this to the Columns using the Classes list in the Columns dialog, which I have just added in JCE Pro 2.9.21 Beta - https://www.joomlacontenteditor.net/downloads/editor/pro/development

Is the Styles list in the editor toolbar also empty?

What items are available in the Classes list in the Columns dialog?

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