Hi, I don;t mind helping you with this at all and to be truthful it does help us all learn how things can look different on different Moodle set ups, so in that respect I am treating this as a learning curve.
You mentioned 'plug-ins'. Now these can be a source of conflict with themes as in some cases they are related in the way that they are made. Blocks especially can do strange things to a theme, simply by the CSS that they use, or rather in the way they use CSS. Some elements will share commonalities, so if a block styles a header h2 in a shade of green, and the theme uses a shade of red, then you could get red headings that should be green and green heading that should be red. Or in your case vertical lists of icon links, where they should be horizontal. So the missing profile images could be one of these 'plug-ins' conflicting with the theme, but that said, this would have to happen whatever the theme. So if this is only happening when this is a user theme, then the possibility here is that the link to the user picture has been misdirected, which means either it was coded incorrectly, or that we may have found a Moodle bug, in that Moodle is looking in the wrong directory, which is a core coding error.
This of course is just an hypothesis, mere guesswork on my part, I need to be able to replicate it to prove the correlation. So I will try to do just that, so I'll let you know how I get on.
In the mean time, can you list the plugins that are present on the page when the missing images occur or when/if the facebook /twitter/ etc., icons are still vertically listed