Not sure why this is still open. Could have sworn I already wrote a complete response.
If you are not getting proper output of the image html it’s not this plugin. To clarify we don’t do anything special. We call get_avatar() which is a WP core function.
If it’s rendering a string there is likely only one possibility, your theme has a custom menu rendering function which changes how menus are output. Likely they are escaping all the menu text assuming it’s never gonna be html. In this case it causes what you see.
As for Facebook images, again not the fault of this plugin as we again are using a core WP function to get the avatar image. That generally means your Fb avatar plugin is not always filtering get_avatar and is selectively doing so for whatever reason.
You can confirm this yourself here: https://github.com/JunglePlugins/User-Menus/blob/master/includes/classes/menu/items.php#L147
Both of those are 100% out of the scope of what we can fix. We can only trust that other plugins are properly rendering menus and properly filtering avatars all the time.
Hope that helps.