Why is the neuronal plasma membrane more permeable to potassium ions than sodium ions?
The cell membrane is inherently quite impermeable to any ion, because it is made of hydrophobic fatty acids. Ions are hydrophilic because of their charge and small size. Metal ions are relatively small and, indeed, have therefore a high charge density. They gather a lot of water around them and hence, they are hydrophilic. 

Therefore ions, including Na+ and K+ are transported via dedicated ion channels, which form specialized pores in the membrane. They consist of vast families with many members and a host of different characteristics. Hence, the size of the ion does not matter, because they are transported through channels. 

There are indeed leaky K+ channels, but not leaky Na+ channels. And exactly this feature is the reason cells are negative on the inside, because positively charged K+ leaks from the cell.

There is just so much the Na+,K+-ATPase can do. In the end  it burns ATP to fuel each cycle to pump out Na+ and pump in K+. At a certain point it cannot transport anymore Na+ out (K+ diffuses relatively freely) because the concentration gradient of Na+ and the electrical gradient (inside negative) gets too steep.