Answers
Answering only the first question:
1) TRUE. Inclusion of an additional variable can reduce the risk of omitted variable bias in your multiple regression. But, such variables do not contribute much to the explanatory power of the model. So, the estimators of the other variables are not effected much. These variables tend to reduce the degrees of freedom.
Also take for example, include a third irrelevant variable X3 into the model with a coefficient zero.
Although the estimators will still be consistent but, if the variables X2 and X3 are correlated somehow, then the correlation coefficient may lead to an increased variance of the estimator. Hence, reducing the efficiency of the model.
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