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.

$$\exists x.\lnot p\left(x\right) \not\rightarrow \exists x. \,\,p\left(x\right)\tag{1}$$
$$\exists x.\;\;p(x) \not\rightarrow \exists x. \lnot p(x)\tag{2}$$
$(1)$ Within a given domain (I'll use human beings in my counter-example to your proposed implications), the existence of someone without a property does not imply the existence of someone with the property. 

There exists humans who are not bears. But that does not imply that there exist humans who are bears.

Suppose it's true that all humans who exist are not bears; we can still assert (truthfully) that therefore, there exist humans who are not bears. But it does not follow  that there must therefore exist humans who are bears.

$(2)$ Within a given domain (again, I'll use human beings in a counterexample to your vice  versa "claim"), the existence of some $x$ such that $p(x)$ is true does not imply the existence of an $x$ such that it is not the case that $p(x)$ holds. 

There exist humans who sleep. But there do not exist humans who don't sleep.

