$(a)$ $L(G_1) = \{a^nb^m \mid n \neq m\}$

$(b)$

- $S\rightarrow S_1S_2\mid S_2S_1 \qquad (2)$
- $S_2\rightarrow aS_2b \mid \epsilon \qquad (2)$
- $S_1 \rightarrow \epsilon\qquad (1)$

So, totally $2+2+1 = 5$ extra productions.

$(c)$ If grammar is ambiguous and no unambiguous grammar is possible for the language, then language is inherently ambiguous.

Non-determinism of language (if it means, $L$ is not DCFL) + ambiguity for all possible grammars implies inherent ambiguity. Or if a language is deterministic, it is surely not inherently ambiguous. But if a language is not deterministic, it may or may not be inherently ambiguous.

The given language $L_2$ is the set of strings where number of $a's$ is followed by an equal number of $b's$ and if not, for the remaining part, the number of $a's$ is followed by an equal number of $b's$. This is actually a deterministic language as we do not need to do multiple counts here. Only after the first condition is violated we need to start doing the count for the second part. This makes the language deterministic and hence it cannot be inherently ambiguous.