If my calculations are correct, a counterexample for question 2 is $$f= \frac{720 + 1684x + 1350x^2 + 585x^3 + 90x^4 + 11x^5}{120} \\ g = \frac{600 + 1434x + 1175x^2 + 535x^3 + 85x^4 + 11x^5}{120}$$ having $$\mathscr{W}f= 6 + x + x^2 + x^3 + x^4 + x^5 \\ \mathscr{W}g = 5 + 2x+ x^2+ x^3 + x^4 + x^5 \\ \mathscr{W}(fg)= 30 + 854x + 4320x^2 + 7292x^3 + \color{red}{7194}x^4 + 7442x^5 + 2059x^6 + 962x^7 + 300x^8 + 38^9 + x^{10}$$

If my calculations are correct, a counterexample for question 2 is $$f= \frac{720 + 1684x + 1350x^2 + 585x^3 + 90x^4 + 11x^5}{120} \\ g = \frac{600 + 1434x + 1175x^2 + 535x^3 + 85x^4 + 11x^5}{120}$$ having $$\mathscr{W}f= 6 + x + x^2 + x^3 + x^4 + x^5 \\ \mathscr{W}g = 5 + 2x+ x^2+ x^3 + x^4 + x^5 \\ \mathscr{W}(fg)= 30 + 854x + 4320x^2 + 7292x^3 + \color{red}{7194}x^4 + 7442x^5 + 2059x^6 + 962x^7 + 300x^8 + 38^9 + x^{10}$$

To Sam Hopkins' suggestions in comments, a counterexample for palindromes is $$f = 2 + 4x + 3x^2 + x^3 \\ g = 2 + 3x + 3x^2 \\ \mathscr{W}f = \mathscr{W}g = 2 + 2x + 2x^2 \\ \mathscr{W}(fg) = 4 + 56x + 180x^2 + 104x^3 + 16x^4$$

And a counter-example for $\gamma$-non-negative is $$f = \frac{6 + 16x + 15x^2 + 5x^3}6 \\ g = \frac{2 + 5x + 5x^2}2 \\ \mathscr{W}f = \mathscr{W}g = 1 + 3x + x^2 \\ \mathscr{W}(fg) = 1 + 36x + 131x^2 + 76x^3 + 6x^4$$