Should the distribution of the residuals of a regression model be symmetric about 0? Basically, should any decent regression model overestimate 50% of the time and underestimate 50% of the time (in the limit)?
In a scenario where a regression model outputs a price, which has to be non-negative, would the distribution of the residuals of an unbiased model still be expected to be ~symmetric around 0? 
Or does the asymmetry in this price example mean that we expect the residual distribution to have a positive skew, since there's no limit to how much the model can overestimate but there is a limit to how much it can underestimate (due to the non-negativity)? Or does it depend on exactly what model is being used and the assumptions the model makes about its error terms? 
If a positively skewed residual distribution is expected, is a simple regression adjustment of the predicted values a viable way to improve model accuracy?
 A: A skewed residuals distribution would imply that your model is biased and keeps over or under estimating. So i think the short answer to your question is yes.
In you example scenario your model would predict the conditional mean price. You want to avoid your model systematically over/under estimating the price which means that the residuals are ideally white noise / normal random variable. 
A: For ordinary linear regression distribution of the residuals of a regression model should be symmetric about 0 since they should have a $N(0,\sigma^2(I-H))$ 
distribution.
Even the price is non-negative, the residuals still could be symmetric around $0$ since the structural component of the model i.e $Xb$ can be anything (include negative values). So using the ordinary least regression might not cause any bias.
However, if your residual is not symmetric about 0 then it suggests you might not use ordinary linear regression. You may consider generalized linear regressions. 
The key is residuals not the value of price(i.e only be positive).
