Step by step working of PN junction diode
For your first question; 
The diffusion currents of minority carriers (electrons in the P side, and holes in the N side) won't affect the overall polarity of each side of the junction very much. Electrons diffusing to the P-side, for example, can be replaced by the negative terminal of the battery and holes diffusing to the N-side can be replaced by the positive terminal of the battery. Thus the battery maintains the voltage across it.

As for your second;
If the physical length of the diode is too short, the probability that an electron or hole will recombine is very low. The recombination of a given electron-hole pair is essentially random but particular diode structures can affect the probability. This is actually a major issue in the design of LEDs as radiative recombination is necessary for their operation. Typically, more complicated structures are used to ensure recombination rates are high such as Quantum Wells and Heterojunctions.