For Facebook, you have to register a personal page first. You can use the privacy settings to make your personal FB page not appear in search engines, or on FB search. You don't have to do anything with your personal page and no one has to know it exists. FB will bug you to fill in personal info, but you are not required to do so.
Using your personal FB page, you then set up your business/art page. No one will know that you are the administrator of your business page. You can remain completely anonymous.
You can put a FB "Like" button on your business website. The Likes go to your Facebook page, not your business website. You can use FB to build brand awareness and to draw people to your website.
Social media can help with creating backlinks to your website, which sends a signal to Google that you must be of interest. This helps boost your website rank in search results.
Social signals were thought to be part of the complex algorithm that Google uses to rank search results. Google's spam guru recently made the subject clear as mud:
Google's Matt Cutts: We Don't Use Twitter Or Facebook Social Signals To Rank Pages
Google is thought to be giving Google Plus more attention. Google Plus is confusing, not as straightforward as Facebook to me. I find it more difficult to make connections, but I do think it is helpful with ranking in Google search results.
People can +1 your business site or your Google Plus page. I'll try to find an article to link that explains how to set it up and best practices.