Is this a valid logical paradox? In some recent cases, I have noticed some theorems are accepted to be intuitively or logically true if they themselves, as a unit, have no valid proof, but, their statements can be used to prove another theorem which then, logically, conversely proves the theorem with no proof. And, I was just wondering if this is actually a valid and logical system. To rephrase it more lucidly: 

"If theorem $p$ cannot be proved, but, can be used to prove theorem $q$, can $q$ prove $p$, and is $p$ thus provable?"

 A: It seems like there's a confusion with the term "provable" here.
Provable means "can be proved". But proofs do not come out of thin air. Sure, you can talk about tautologies and logically valid statements, but those are quite boring and uninteresting.
Proofs begin with axioms. And axioms are simply assumptions. Things which can be proved from our axioms are said to be provable, and usually we say that $\varphi$ is a theorem of $T$ if it is provable from $T$.
For example Zorn's lemma is unprovable from the axioms of $\sf ZF$, but it is a theorem of $\sf ZFC$, and in fact the axiom of choice (the $\sf C$ in $\sf ZFC$) is a theorem of $\sf ZF+\textrm{Zorn's lemma}$.
But proofs can be almost tautological. $\varphi$ is provable from the theory $\{\varphi\}$ in the most obvious and uninteresting way. So taking any $\varphi$ which $T$ cannot prove, $T\cup\{\varphi\}$ can prove anything that $T$ could, but it doesn't mean that those statements are necessarily unprovable from $T$ itself.
This sort of situation is a key issue in modern set theory. We investigate the provability relations between various statements. We can prove that $2^{\aleph_0}=\aleph_1$ assuming $\lozenge$ holds (and the inverse cannot be proved); and we can assume the continuum hypothesis implies all sort of statements. The fact that the standard axioms of set theory, $\sf ZFC$, are consistent with both the continuum hypothesis and its negation only make things more interesting. But it doesn't mean that whatever you prove by assuming $\sf CH$ is automatically unprovable. Sometimes we work quite hard, and sometimes we even succeed in showing that these sort of assumptions can be removed.
With all that being said and done, of course in the context of a particular mathematical theory you can just say "unprovable", although it is probably better to use "independent" (because something which is provably false is not as interesting as something which is neither provable nor disprovable). So if in a set theory course the teacher says that something is unprovable, there is a good chance they mean "...from $\sf ZFC$". And so statements which are proved using assumptions which are themselves independent can turn up independent themselves, or they can turn up provable after all.
A: No, this is not true.  Given any contradiction you can prove everything from it.  And since we don't want to accept contradictions...
While you didn't provide any examples of the situation you're asking about, my guess is that what you're really talking about is that a theorem will be stated without proof, usually because the proof is too difficult, but the theorem is too useful for the rest of the course to not use it.
A: No, because any statement can be "proved" from a false statement, including the original false statement.
