Fermi level upto which Boltzmann approximation can be made
Bolzmann's approximation means replacing the Fermi-Dirac distribution function by the simpler "exponential only" one (=no denominator needed) which is Bolzmann's statistical distribution function. This approximation is valid when the exponential in the Fermi-Dirac distribution function is large enough when compared to 1. Your book states that large enough is exp(3) or approximately 20.

Your first equation is from where this approximation is used to calculate the needed N doping concentration from the wanted new Fermi level.

The boron example applies it to the acceptor doping Na, because the holes also obey the same distribution law, but everything is mirrored relative to midgap.

The formula for the difference between the doped and intrinsic fermi levels = Eg/2 - (Ea-Ev) - (Ef - Ea) is a clever shuffle to insert the difference of the new fermi level and the new boron-introduced acceptor energy level.