Why does the distribution of the residuals should be normal and variance of the residuals should be same across the values of the independent variable? Why do we assume these for the residuals only? What are the implications of violating those assumptions? Instead of a mere conceptual explanation, could you please give solid examples such as "at this point in linear regression, we assume this assumption and violating this makes these calculations incorrect" etc?
MLS will give the best estimates as intercept and coefficients to predict the dependent variable. The rest is the residuals or the errors. If the fit provided reflects the data accurately, the errors are simply observational errors and will be normally distributed. If they are not, it means that the best linear estimates for intercept and coefficients do not capture the relationship between the dependent and independent variables. The difference is only seen in the residuals because the rest is fixed by the model generated. The distribution of the residuals contains lots of information that can be used to restate the model and maybe obtain a better fit.