Consider the simple linear model $Y = \beta_0 + \beta_1X + \beta_2 X^2$. Now think of a change of scale (in your case, that could be for instance measuring in square feet instead of square meters, or viceversa) so that the new variable is $Z = aX + b$. The very least you would ask of an statistical model is that the inferences are invariant to the scale of measure. See how the model changes when writen in terms of $Z$.
Y &=& \beta_0 + \beta_1Z + \beta_2Z^2 \\
&=& (\beta_0 + \beta_1b + \beta_2b²) + (\beta_1a + 2ab\beta_2)X + a^2\beta_2X^2 \\
&=& \beta_0^* + \beta_1^*X + \beta_2^*X^2
Assume you want to test that the quadratic effect is zero. You can do it in either model, as $\beta_2 = 0$ implies (and is implied by) $\beta_2^* = 0$, since $\beta_2^* = a^2\beta_2$; so the model gives invariant inference about the quadratic term in either scale of measure when you include the linear term. This is no longer the case if you do not include the linear term, as can be checked easily.
So to answer your question, yes, you want to include the linear term even if interest focuses on the quadratic term, as otherwise you might get different answers in different scales. Or, put in another way, you want to fit a well structured hierarchical model.
Notice this reasoning also underlies the inclusion of an intercept, even if per se the intercept terms is of no interest.
Regarding your last question, fit the model that seems best and then obtain the first derivative, as has been advised in a comment.