Why are errors independent but residuals dependent?
As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also we assume that $\mathbb E(\epsilon)=0$. Why doesn't it imply errors are also not independent?
On
Your question is best explained in a broader context: What is the difference between "error" and "residual?"
In regression, residuals are calculated based on a fitted model for which the underlying parameters are estimated from the data we observed, because those underlying parameters are unknown to us. For this reason, residuals are not independent: a constraint is imposed on the model fit to make the estimated parameters uniquely determined (as in the case of ordinary least squares fitting in linear regression).
This speaks to a subtle but important property of residuals: they are in a sense estimates or realizations of error conditional on the assumption that the true error is faithfully represented by the data you observed. Error in a model is intended to capture natural random variation of the response (dependent) variable not explained by the predictors (independent variables). But a residual could be calculated from any model fit and it need not be true to this underlying error.
On
The simplest case is this:
$X_1,\ldots,X_n$ are uncorrelated and have expected value $\mu$ and variance $\sigma^2.$
$\overline X = (X_1+\cdots+X_n)/n$ therefore has expected value $\mu$ and variance $\sigma^2/n.$
The errors are $\varepsilon_i = X_i-\mu.$ The sum of the errors has variance $\sigma^2/n,$ so the sum is not zero.
The residuals are $\widehat{\varepsilon\,}_i = X_i - \overline X.$ The sum of these is necessarily zero, so these are negatively correlated.
Your model can be biased, hence the errors can sum to something other than 0, while the residuals, as you correctly point out, are constrained.