Can't advise on the space, looks like it's coming from pre-text or greeting? Not sure, you've the module set as horizontal, but your design is vertical. I recommend reconfiguring within the modules parameters in module management. Without a direct link I can't see source and what is causing it.
For social media links we're using the below extension with custom icons.
extensions.joomla.org/extensions/social-web/social-profiles/11989 
krileon
