Why don't the minority carriers combine with the majority carriers in a PN junction?
The minority carriers in a semiconductor are not from manufacturing defects, but from spontaneous generation. Basically, in a semiconductor, electron-hold pairs are continuously being generated (using thermal energy). At equilibrium, these are also recombining when they (later) run into each other. A balance is created with the number or carriers being a strongly increasing function of temperature. 

Now, when the semiconductor is doped, the number of majority carriers will be very high, and so when minority carriers get spontaneously generated (which still occurs), they have a much higher chance of recombining with a majority carrier quickly. This means that the number of minority carriers will decrease, but will not become zero.

There is an equation that describes this (Law of Mass Action). Basically, the product of Majority*Minority carriers is a constant (at a given temperature). When doped, the Majority increases and the minority decrease.

Undoped, this gives ne = nh; ne x nh = constant. When doped to Ne, you get ne x nh = Ne x Nh.