Question 1131226
I will answer only 1 and 3.  Number 2 is an engineering/legal issue, not a statistical one.
for men, want a z<=(56.3-69.9)/3.7 or -13.6/3.7 which is -3.67.  Nearly no man will fit through without bending, the percentage being 0.01%
for women it is z<=(56.3-63.6)/3.7=-7.3/3.7=-1.97
that probability is 2.44%

z.40 is at -0.25
-0.25=(x-69.9)/3.7
-0.925=x-69.9
x=68.975 inches or about 69 inches