This is a nonstatistician's stab at your questions:
First the word assumed means something a bit different than you may think it means. It means that in order for this test to be appropriate the data must fit this specification. So one of the assumptions of the test you are using is that data comes from a population that follows the normal distribution (notice I didn't say the data is normally distributed as this is rarely the case). If the errors are normally distributed then this is an indication that the data comes from a normally distributed population.why is it assumed that the errors e_i are normal distributed in linear regression, see e.g. Seber.
According to Cohen, Cohen, Aiken and West (2003) the violation of normally distributed error terms does not lead to biased estimates. "The affect of violation of the normality assumption on significance tests and confidence intervals depends on the sample size, with problems occurring in small samples."What would be the influence e.g. on confidence intervals for parameter estimates, if one neglect this assumption