Big O /Right or wrong?

You can't go from $f \leqslant c_1 h$ and $g \leqslant c_2 h$ to $\frac{f}{g} = \frac{c_1}{c_2}$.

And the initial claim is false. Take, for example, $f(n) = h(n) = n^2$, $g(n) = n$.

Consider $f(n)=h(n)=1$ and $g(n)=1/n$.