1) The polynomials $x^2 + x + 100$, $x^2 + x + 100$, $-x^2 + x - 200$, $1$ give a needed example.

2) To the contrary, assume that there exists a special set of polynomials $f_1, f_2, f_3, f_4, f_5$ with real coefficients. Consider a complete graph with a set of vertices $\{1, 2, 3, 4\}$. An edge $(i, j)$ is called positive (negative), if the polynomial $\frac{2}{3}f_5 + f_i + f_j$ is positive (negative). By the assumption in this graph there is no any positive (negative) triangles. Indeed, if for example $\frac{2}{3}f_5 + f + g > 0$, $\frac{2}{3}f_5 + g + h > 0$ and $\frac{2}{3}f_5 + f + h > 0$, then $f_5 + f + g + h > 0$ which is a contradiction with the definition of a special set. The next lemma can be easily proved.

Lemma. In a two-colored complete graph with four vertices without one-colored triangles there is a pair of skew (crossed) edges of every color.

Using the lemma we have conflicting inequalities:
$$f_1 + f_2 + f_3 + f_4 + \frac{4}{3}f_5 > 0,$$
$$f_1 + f_2 + f_3 + f_4 + \frac{4}{3}f_5 < 0.$$