questions about sensitivity in differential privacy
$D$ is the database at hand. Differential Privacy is a property of a randomized algorithm, which is associated with a domain. So, DP holds for any neighboring databases in the domain. However, when you have a true database $D$, in some cases you don't need to add a noise whose distribution is associated with the global sensitivity to ensure privacy. You can add a smoothened version of the  local sensitivity to add a smaller noise, where local sensitivity is roughly the sensitivity of the function restricted to the neighbors of $D$.