Expected value of average of error-term I'm having some trouble understanding the following expression:
$E(\bar{u}|x_i)=0$
$u$ is the error term and $\bar{u}=n^{-1}\sum u$
It's used for proving unbiasedness of OLS.
I've got two questions:
Does $\bar{u}$ equal zero like for the residuals? I know that the expected value of u equals zero.
If yes, then taking expectation of $\bar{u}$ is like taking expectation of a constant and therefore the expected value is redundant?
 A: The main issue here is that you are confused. Let us look at this in the setting of simple linear regression, the model is
$$
    Y_i=\beta_0 + \beta_1 x_i+u_i, \quad i=1,\dotsc n
$$ (You didn't write a model equation in the question, doing that is often a first step in clearing things up.)  For instance, you write only $u$ and not the more correct and informative $u_i$. Under the usual OLS assumptions we have $u_i$ are iid (independent identically distributed) with mean 0 and variance $\sigma^2$. Then we have 
$$ \DeclareMathOperator{\E}{\mathbb{E}}
   \E \bar{u}=\E \frac1{n}\sum_1^n u_i =\frac1{n} \sum_1^n \E u_i =0
$$
But then in a comment you say:
If we'd know the true regression line, wouldn't the $∑u_i$ equal to zero?
But, even if you know the true values of $\beta_0, \beta_1$, the error terms $u_i$ are still independent random variables as above, and there is no reason their sum is identically zero. That is not consistent with independence (and positive variance.)
It is not clear what you mean by data of the whole population so I will not comment on that part.
