When developing a CFG, I find that one can be confused about whether the grammar is correct, i.e. whether it recognizes only the required strings and not other strings.
But this can be hard to see?
Are there techniques and tools for "proof-reading", whether the grammar is correct and succinct enough?
There is no general method for your problem. Indeed the problem whether a CFG accepts all strings is undecidable. However you can always try to find a formal proof.
Assume you want to prove that the grammar $G$ $$ S\to aSb \mid \varepsilon$$ generates the language $L= \{a^nb^n \mid n\ge0\}$. Then you have to show that (1) every string $a^kb^k\in L$ can be derived from $G$, and (2) every $w \in L(G)$ has the form $a^kb^k$.
Both statements are not difficult to prove. For example to show (1) you can argue as follows. If we apply the $S \to aSb$ rule $k$ times, then we get the string $a^kSb^k$. After substituting $S$ with $\varepsilon$ we obtain $a^kb^k$.
