Hello Aliamac,
To quickly introduce myself, I am with Core Techies who are functioning from last 6+ years, having expert developers in Drupal PHP Website Development & Customization. I can do all three points :
1) Add a narrow fixed header to run across the top of the website in which to place the Log In button as well as the social media icons so that they are always visible. The header will need look good in a desktop, tablet as well as phone screen format.
2) Move the Log In button to the new fixed header and build recognition of the user's name, eg. Log in changes to "Hi Alex", when they are logged in.
3) Move the social media icons (provided) to the new fixed header
for you in a very short time and low budget. Please initate the chat to discuss the requirement and start working on it.