My attempt:
a) $g(t)=t^3*\vec{c}$ curve is parallel to $\vec{c}$ because $\vec{c}\times t^3*\vec{c}=[0,0,0]$.
b)$h(t)=e^t*\vec{c}$ curve is also parallel to $\vec{c}$ because $\vec{c}\times e^t*\vec{c}=[0,0,0]$
c) My answer to part (c) is f'(0) and g'(0) are both zero vectors. What would be your answer to part (c)?
If any member knows the correct answer to this question (c) may reply with correct answer. I think my answers to part (a) and (b) are correct.
Your answers to parts a and b are correct, although your reasoning isn't entirely. Cross products are only defined for vectors in $\mathbb{R}^3$, so your reasoning doesn't apply in general.
Your answer to part c isn't correct.
All of these functions, as you said, are lines parallel to $\mathbf{c}$. Note that $\mathbf{c}$ is constant with time. So, we have
$$ f'(t) = \mathbf{c}$$
$$ g'(t) = 3t^2\mathbf{c}$$
Although the functions don't have the same derivatives, their derivatives are again parallel. Imagine you have an infinite line parallel to $\mathbf{c}$, and a point somewhere on this line. As time passes, the point with position defined by $f$ travels much slower than that defined by $g$.
EDIT: Sorry, I misread part c, I didn't realize it only asked to compare $f'(0)$ and $g'(0)$. That is a little weird, I wonder if they meant $g'(0),h'(0)$.