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For nontrivial groups $A,B$, the free product $A\ast B$ is amenable exactly in the exceptional case $A\cong B\cong C_2$. In that case it is the infinite dihedral group, hence a virtually cyclic group and amenable. In every other case the normal form theorem for a free product and the ping-pong lemma produce a subgroup isomorphic to $F_2$; the fact that a subgroup of an amenable group is amenable then obstructs amenability.

Yes, but there is only one nontrivial isomorphism type that works. If $G=C_2$, thenThe infinite dihedral group contains its infinite cyclic rotation subgroup with index two, so it is amenable by virtually abelian groups are amenable.

Suppose now that $G\ncong C_2$. The action of $G\ast G$ on its Bass-Serre tree is non-elementary: one vertex degree is greater than two, so there are hyperbolic elements with disjoint pairs of endpoints. Suitable powers satisfy the ping-pong lemma on the boundary and generate a copy of $F_2$. Since a subgroup of an amenable group is amenable, an amenable group cannot contain this nonamenable subgroup. The amenability of a free product therefore yieldsfor nontrivial finitely generated $G$.