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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 $$\mathscr{W}f= 6 + x + x^2 + x^3 + x^4 + x^5 \\ \mathscr{W}g = 5 + 2x+ x^2+ x^3 + x^4 + x^5 \\ \mathscr{W}(fg)= 30 + 854x + 4320x^2 + 7292x^3 + \color{red}{7194}x^4 + 7442x^5 + 2059x^6 + 962x^7 + 300x^8 + 38^9 + x^{10}$$