Form of a monotonic decreasing function
No, those first-order informations are not enough to conclude anything like that.
As a counterexample consider the function $y:\mathbb{R} \to \mathbb{R}$ defined by $$ y(x) = \lvert x \rvert^{3/2} $$ Then $y$ satisfies all of your conditions but is not in $O(x^2)$ as $x \to 0$. If you're interested in the asymptotics for $x \to \infty$ then there's also multiple easy to find counterexamples.