I would like to be sure whether my predicate logic formulas are consistent with given statements, since I struggle in such tasks. Would appreciate your thoughts about it:
$A(x)$: $x$ is an animal;
$G(x)$: $x$ is green;
$E(x)$: $x$ is an elephant;
If an animal is green, it is an elephant: $$\forall x((A(x)\wedge G(x)) \rightarrow E(x))$$
No green animal is an elephant: $$\forall x((A(x)\wedge G(x)) \rightarrow \lnot E(x))$$
Yes, that looks good. Since you are able to express these sentences accurately, now might be a good time to start thinking about how to express statements made with the universal quantifier in terms of the existential quantifier.
For example, you could also express "No green animal is an elephant" with the following expression:
$\neg \exists x(A(x)\wedge G(x) \wedge E(x))$
since $\forall x(M(x)$ is equivalent to $\neg \exists x(\neg(M(x))$ where M(x) = [$(A(x) \wedge G(x)) \rightarrow E(x)$] could be used.