I'd quite like to make the category image and the container it is in take up less space.

I think it is called .codo_categories_container

Can I alter its width?

Could look a bit awkward of different devices??

The forum is build on grid using bootstrap, so do not alter the width using css.

To alter the width, open

sites/default/themes/default/templates/forum/topics.tpl

Then

Search for div

 <div class="codo_categories col-md-4" id="codo_categories_sidebar">

Reduce col-md-4 to col-md-3 or less

Then find

<div class="codo_topics col-md-8 clearfix">

Increase col-md-8.

The numbers when added should sum up to 12 (columns)

But, as @hard_as_snails said, it may not look good.

Terrific, thank you. I might experiment with different sizes.

What about scaling as a percentage of screen width, rather than an absolute value. Can I make the sidebar, say, 20%, and the main posts 80%?

They are defined in % when you use those col-md-* classes.