Inference about the true intercept of the model and the OLS being BLUE Consider the following population regression model:
$$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$
where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = \alpha \epsilon$ for $0\leq \epsilon \leq 1$.
I have the following data: $X = (3,1,2,2)^{'}$, and $Y = (3,0,1,4)^{'}$.
By calculating the estimated intercept and the slope, I have $b_{1} = -1$, and $b_{2}=1.5$.
Two classical assumptions are violated in this model: one is no mean-zero errors $E(\epsilon {_i})\neq 0$, and the other one is the errors not being normal. 
From this, can I infer anything about the true intercept of the model? I think we could either be overestimating or underestimating the true intercept. 
Also, as long as I have an intercept in the model, then is the OLS still BLUE? I think that our estimate of the intercept is biased, but the estimate for $\beta_{2}$ is still BLUE because we have an intercept in the model. 

Note the Following. 
Let $\bar{e}$ denote the mean of e. Then I can re-write your model as 
$y_i = (\beta_1 + \bar{e}) + \beta_2x_i + (e_i - \bar{e})$. 
This model is identical to yours and now has a mean-zero error term (though, as you point out, it is still not normally distributed), but the intercept will be "biased" by the mean of the original error.
