Category logo on all topics within this category

We remove the logo:

img#site-logo {
    display: none;
}

Or maybe this:

.category-support img#site-logo {
    display: none;  
}

I added for clarity your photo:

.category-support .title a {
    background-image: url(https://d3bpeqsaub0i6y.cloudfront.net/user_avatar/meta.discourse.org/ed_bobkov/45/177564_2.png);
    width: 35px;
    height: 35px;
    display: block !important;
}

You may have to repeat this code 30 times.

I gave an example of how to do this without a component / plugin. You can implement it somehow differently. Although, css for 30 categories does not take up much space.