= Solution
Let $\phi=2\cos(\pi/5)=(1+\sqrt5)/2$. In the vertex order along the displayed $3$-$5$-$3$ path, the <Coxeter Gram matrix> has diagonal entries $2$ and successive off-diagonal entries $-1,-\phi,-1$. Its leading principal determinants satisfy
$$
D_1=2,qquad D_2=3,qquad D_3=6-2\phi^2=4-2\phi,qquad D_4=2D_3-D_2=5-4\phi<0.
$$
The last determinant is nonzero, so the form is nondegenerate. Its negative determinant rules out positive semidefiniteness and hence also positive definiteness. Thus the answers are respectively no, no, and yes.
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