Adjusting the display of In "Category".

That’s not possible at the moment but we’ll add 2 new filters, jetpack_relatedposts_post_category_context and jetpack_relatedposts_post_tag_context, that will allow you to customize the post context, in the next Jetpack release.

You can follow the conversation that lead to the addition of the filters here:
https://github.com/Automattic/jetpack/issues/361

Thanks for the response! That will be fabulous. Also, is there a way to change the layout of the post image?

Yes, you can use the jetpack_relatedposts_filter_thumbnail_size filter to change the size of the images:
https://github.com/Automattic/jetpack/blob/3.1.1/modules/related-posts/jetpack-related-posts.php#L729

You can also use CSS to change the layout itself.

Great! I’m a bit new to customizing functions, so can you give me an example of what the string would look like?

Thanks again!

That thread includes some examples that should help you:
https://www.ads-software.com/support/topic/change-thumbnail-size-for-related-posts?replies=9