What's the bias of random polynomials with low degree over GF(2)?
Your question is equivalent to tail bounds on the weight distribution of Reed-Muller codes. Understanding weight distribution of Reed-Muller codes is an old and challenging question in coding theory, and several interesting results are known about it (the weight distribution is completely understood only for $d=1$ and $d=2$). As a great starting point, see "Weight Distribution and List-Decoding Size of Reed-Muller Codes" by Tali Kaufman, Shachar Lovett, Ely Porat, and the references therein.