Is $\;\exists x.\lnot p\left(x\right) \rightarrow \exists x. p\left(x\right),\;$ and vice versa, always true? I have a feeling this could be true for all cases, for example when I state that some fruits are not apples, does not this automatically mean that some fruits are (and vice versa)? That is, is it true that:
$$\exists x.\neg p\left(x\right) \to \exists x. p\left(x\right)\quad?$$
And what about: $$\;\exists x. p(x) \rightarrow \exists x.\lnot p(x)\quad?$$
On the other hand, I could not come up with any formal proof so I would like to hear your thoughts as to why I'm right or wrong about this.

Nope. "There exist some oranges that are not apples" is a perfectly true statement, whereas "there exist some oranges that are apples" is perfectly false. 
