The answer to both questions is yes. For the first, assume only that $f \in C^2$, that $f(0) = f'(0) = 0$, and that $|f''| \leq M|f|$. Let $g(x) = f^2(x) + f'^2(x)$. Then $$g' = 2f'(f''+f) \leq 2(M+1)|ff'| \leq (M+1)g.$$ Hence $(e^{-(M+1)x}g)' \leq 0$, and since $g \geq 0$ and $g(0) = 0$ we conclude that $g = 0$.

This is a very simple form of Carleman estimates, which can be used to show that if $|\Delta f| \leq M|f|$ and $f$ vanishes to infinite order at a point, then in vanishes in a neighborhood of this point. An excellent reference for this is a set of lecture notes by Carlos Kenig, which can be found here.

The answer to both questions is yes. For the first, assume only that $f \in C^2$, that $f(0) = f'(0) = 0$, and that $|f''| \leq M|f|$. Let $g(x) = f^2(x) + f'^2(x)$. Then $$g' = 2f'(f''+f) \leq 2(M+1)|ff'| \leq (M+1)g.$$ Hence $(e^{-(M+1)x}g)' \leq 0$, and since $g \geq 0$ and $g(0) = 0$ we conclude that $g = 0$.

This is a very simple form of Carleman estimates, which can be used to show that if $|\Delta f| \leq M|f|$ and $f$ vanishes to infinite order at a point, then $f$ vanishes in a neighborhood of this point. An excellent reference for this is a set of lecture notes by Carlos Kenig, which can be found here.