I'm just new to regression. I would like to know if I'm having a linear model. Does unbiasedness breakdown if i fail to account for heteroskedasticity?
The answer to your question is yes. If you apply ordinary least squares to obtain the parameter estimates of the regression coefficients in linear regression, estimates will all still be unbiased if all assumptions hold other than homoscedasticity. Heteroscedasticity causes model misspecification and can hurt predictions if not accounted for. But in the face of heteroscedasticity the least squares estimates remain unbiased. You can find this on wikipedia under the title of heteroscedasticity. A search of this site should turn up information on this also. You should check for possible duplicates to your question.
Another point about OLS, the Gauss-Markov theorem states that the least squares estimates of the regression parameters are the best linear unbiased estimates even when the error terms are correlated but the model is still homoscedastic.