Part 1

The main trick is to multiply the first equation through by 4 which results in:

$4(a^2 + b^2 + c^2 + d^2 + e^2) = 4a(b + c + d + e);$

or $(a^2 - 4ab + 4b^2) + (a^2 - 4ac + 4c^2) + (a^2 - 4ad + 4d^2) + (a^2 - 4ae + 4e^2) = 0;$

or $(a - 2b)^2 + (a - 2c)^2 + (a - 2d)^2 + (a - 2e)^2 = 0;$

or $b = c = d = e = \frac{a}{2}.$

If we now substitute these values into the second linear equation, we now obtain:

$2011a + (\frac{a}{2})(2012 + 4026 + 2014 + 4030) = 16232832 \Rightarrow a = 2016;$

and our final calculation comes to:

$\frac{a + b + c + d + e}{9} = \frac{a + 4(\frac{a}{2})}{9} = \frac{a}{3} = \frac{2016}{3} = \boxed{672}.$