No, $F$ is not onto in general. For example, it is not difficult to show that, when $n=3$, the polynomial $$ c = c_0 + c_1t + c_2 t + c_3 t^3 + c_4 t^4 $$ is in the image of $F$ if and only if $$ 12\,c_0c_4 - 3\,c_1c_3 + {c_2}^2 \ge 0. $$

No, $F$ is not onto in general. For example, it is not difficult to show that, when $n=3$, the polynomial $$ c = c_0 + c_1t + c_2 t + c_3 t^3 + c_4 t^4 $$ is in the image of $F$ if and only if $$ 12\,c_0c_4 - 3\,c_1c_3 + {c_2}^2 \ge 0. $$

To see this, note that if $c = F(a,b)$ as above, then $$ 12\,c_0c_4 - 3\,c_1c_3 + {c_2}^2 = (3a_0b_3-3a_3b_0-a_1b_2+a_2b_1)^2\ge0. $$ To prove the converse direction, note that, if the above inequality holds, then, taking the square root of the above equation, one finds that one can solve for all of the expressions $a_ib_j-a_jb_i$ for $0\le i < j \le 3$, and this determines $a$ and $b$ uniquely up to a unimodular linear combination.