Why is the definition of the mole as it is?
Why is the definition of the moles as it is?


It is a rather arbitary definition that the mole is the number of atoms in 12g of carbon 12.  This has not always been the definition.  For example, prior to 1960, the definition was based upon oxygen rather than carbon-12. 

The first standard was based upon 1 gram of hydrogen.  Later, the standard was changed to 16 grams of oxygen being a mole, for convenience because oxygen formed compounds with many other elements.  Eventually it was realized that elements including oxygen have different isotopes of different mass, and the mass of a particular isotope is a more specific and measurable standard than the mass of natural abundance oxygen.  12 grams of carbon-12 matches the old standard of 16 grams of natural abundance oxygen more closely than choosing 16 grams of oxygen 16.  


  can we really neglect the mass of electrons, when the atoms get bigger?


The mass of a proton or neutron is about 1836 times that of an electron. So depending upon the ratio of neutrons to protons, electrons are at most 1/1836 of the mass of a neutral atom.  If accuracy of more than 1 part in 1836 is desired, it is important to consider the electrons, regardless of whether atoms are big or small.    


  why do the element's mass per mole do very rarely end up being integers?


No element other than carbon-12 will exactly be an interger.  This is because masses of atoms depend upon number of protons, neutrons, electrons and binding energy.

In other words, considering that carbon-12 has an equal number of protons and neutrons (and electrons), other isotopes with a 1:1 proton/neutron ratio would have essentially integral atomic masses except for binding energy.  For example cadmium 112 is 111.90 instead of exactly 112 because binding energy is slightly more than 0.1 amu, partially offest by a higher number of neutrons than protons, the neutron mass being slightly greater than the combined mass of an electron and proton.  

As pointed out by Matt Black, the natural abundance atomic masses further deviate because they are weighted averages of the masses of isotopes of a given element, weighted by their abundance on Earth.