Why should/shouldn't I put social media icons on my website?
Putting or not social media icons is mainly a matter of marketing. It strongly depends on the kind of website you're running and your target audience (selling dental implants to elder people might not get that much more traffic from social sites), basically they are a way to attract the more people you can. It's free advertisement, more or less. And, imho, social media icons are quite a 'web 2.0' trend, so you might get benefits from a graphical point of view, and their interactiveness can be a plus for your public. Again, it's 'marketing'.