Is the estimator of the residual e_hat normally distributed if the errorterm e is normally distributed? Y=b0+b1*x+e thnx in advance

If your linear regression model is properly specified (i.e. no assumption violation), e_hat should be normally distributed with an expected value of 0.

of course… had a bad moment when i wrote this question. but all the e_hats are together not independant :-), like the error terms are

If the e_hats are not independent, you have an autocorrelation problem. Run D-W test to be sure and correct it with Hasen method.

thats funny: e_hats are not independant, even if the error terms are independant do a monte carlo in excel and you will see…