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.
 A: $$\exists x.\lnot p\left(x\right) \not\rightarrow \exists x. \,\,p\left(x\right)\tag{1}$$
$$\exists x.\;\;p(x) \not\rightarrow \exists x. \lnot p(x)\tag{2}$$
$(1)$ 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.

A: 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. 
A: Definitely not! Let $p(x)$ be any assertion which is always false, such as $\lnot(x=x)$. 
And the vice-versa part is not correct either, for basically the same reason. 
A: Neither is always true. Some fruits are not apples would still be true even if no fruits were apples. For a mathematical example, let $p(x)$ mean ‘$x$ is an integer, and $x$ is both odd and even’. Then $\exists x\big(\neg p(x)\big)$ is certainly true, since $\neg p(1)$ is true. However, $\exists x\big(p(x)\big)$ is certainly not true, since $\forall x\big(\neg p(x)\big)$ is true.
A: Say that $p(x)$ means "$x$ is an invisible pink unicorn".  Your claim says that if there is an $x$ such that $x$ is not an invisible pink unicorn, then there is an $x$ such that $x$ is an invisible pink unicorn.
My left boot is not an invisible pink unicorn.  Therefore, if your formula were correct, the existence of my boot would be a proof of the existence of invisible pink unicorns.
If your formula were correct, my boot would also be proof of the existence of omnipotent gods, of space aliens, of holes to the center of the Earth, and of flying spaghetti monsters, since it isn't any of those things either.
A: As others have pointed out, this does not follow.  ... But why does it seem to follow? I believe it is because when in real life we use 'some', we very often mean 'some, but not all', otherwise we would have used 'all'. So, in real life, when I say 'some students are good at logic', it is pretty much understood that I also think that some students are not good at logic.  Logic, however, does not care about these natural language conventions.
