How to create sticky Social Media Tabs to the right of the website?
Try Social media module, it will display as block and then assign it to the proper region and AFAIK no module exist as such, you need to customize through CSS.


  The social media module helps integrate your website with social media
  sites such as Twitter, Facebook and Google+. It provides an
  centralized way of managing social media profile information and
  plug-in widgets such as follow and share buttons.


if you are looking to share the posts then Share Bar is the right choice for you.


  The Drupal Sharebar module lets you easily add a "floating" social
  media submission bar on your website (like that found on Mashable)
  that includes share buttons for the most popular social media websites
  online such as Facebook, Twitter, and Google +1 Buttons.