Why does adding a quadratic term to a regression change unrelated coefficients?
Your errors aren't the same anymore.  For example, instead of writing $Y = \beta_1 + \beta_2 X + U$, you're actually writing  $Y = \alpha_1 + \alpha_2 X + \alpha_3 X^2 + V$.  There is no expectation that they should be the same.

In other areas, the trouble is that the error term is likely correlated with your regressors.

Fear not: running wages on education is a common specification.  Searching for something like 


  wages education endogeneity


should bring up a whole slew of papers that treat the topic.