Fitting a polynomial with arbitrary derivative values Given two points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_2 > x_1$ and $y_2 < y_1$, I can obviously fit a line (order $1$ polynomial) to them. But if I want to fit a quadratic function by specifying real, finite values of the derivatives at the two points, are there any constraints on the values of the derivatives?

You have four variables: $y_1$ and $y_2$, and the slopes $m_1$ and $m_2$. This means that you would need a cubic, which has four coefficients: $a_3x^3+a_2x^2+a_1x+a_0$.
Alternatively, if you really must have a quadratic, you could reduce the degrees of freedom by specifying only one of $m_1$, $m_2$, or by requiring that $m_1 = m_2$, or by specifying both slopes but only one of $y_1, y_2$, etc. This would give you three degrees of freedom corresponding to the three coefficients in $a_2x^2+a_1x+a_0$.
