[ShockWave]

Take a look at this one. 

Are you on 4.0.12? If so, a known bug is present in that version that makes it impossible to save a group's settings once they've been changed. 
In order to get this one fixed, you would need to upload a patch that fixes the issue. Unfortunately, said patch doesn't seem to be available on IPS's community site anymore.
Since that's the case, you're gonna want to simply upgrade to 4.0.13(.1) and just have it fixed for good. 

I believe it's this one.
https://community.invisionpower.com/files/file/7615-donations/
 

Try uninstalling all third party themes you might have installed. That's precisely what happened to me as well.

There's now a setting for that in each forum's edit dialog. From within your ACP, all you have to do is find the forum whose icon you wish to change, and click "Edit". An option labeled "Forum icon" is somewhere in the page that loads up next. 

Have you tried restarting your webserver and cleaning your board's cache through Support -> Something isn't working properly? 

The answer provided in this one could very likely be it:

 

{ Head over to your Admin CP - Customization - Themes - (Your preferred theme) - Manage HTML and CSS - CSS (Second tab) - custom.css }. Add the following to it.

body {
    background-color: yourcolor;
} [/code] 

Here you go. 
body {     background-image:url("img_tree.png");     background-repeat: no-repeat; } Give that a try. Upload the image you wish to use to an image service site such as imgur.com, and replace the img_tree.png bit with the address link you get once the image's been uploaded. You can just add that code to your custom.css file, located at your ACP > Customisation > Themes > Edit HTML & CSS > CSS Tab > custom.css. 
I'd give you the exact fragment of code I used to use on my board, but I can't access it right now. Nevertheless, that should work. 

Check the file and line highlighted for any database errors / false data. I do believe this is due to it. Check it out through your cPanel. Last time this happened to me was after having completely deleted my database. (Don't ask..) 

Head over to ipb_styles.css, and look for something like the following:

You can now try messing with the background value.

Try enabling 'Designer's Mode' under the 'Themes' tab in your ACP, and then disable it, while keeping the 'Synchronize Changes' option enabled.

Ah, you did mention you are talking about IPB 3.x. I have to learn to read them tags! This is indeed for IPB 4.x, as I got confused by the example websites you provided, which are using that. Let me look around and see what I can find on IPB 3.
Update: There does not appear to be anything like this for IPB 3.x, besides the all-time classic 'Moderating Team' area, to which we're all used. The following screenshot pretty much confirms so.