About 1., Gödel's First Incompleteness Theorem is a ingenious exercise of "coding" formal properties and relations regarding a theory $F$ with "a certain amount" of arithmetic inside $F$ itself.
This exercise ends with the definition of the so-called provability predicate $Prov_F(x)$ which holds of $a$ iff there is a proof in $F$ of the formula $A$ with "code" $a$.
To complete the proof, [it is used] the negated provability predicate $¬Prov_F(x)$: this gives a sentence $G_F$ such that
$⊢_F G_F \leftrightarrow ¬Prov_F(\ulcorner G_F \urcorner)$ [where $\ulcorner x \urcorner$ is the "code" of formula $x$].
Thus, it can be shown, even inside $F$, that $G_F$ is true if and only if it is not provable in $F$.
Thus, "reading" the above proof, we can "know of" the truth of $G_F$ (provided that $F$ is consistent) simply because $G_F$ is not provable in $F$ and $G_F$ is equivalent to the formula $¬Prov_F(\ulcorner G_F \urcorner)$.
About 2. :
Why doesn't the fact that $G$ is true constitute a proof of $G$ ?
Because a proof in $F$ of $G_F$ is a precise formal objcet and G's Incompleteness Th shows that such a proof in $F$ cannot exists.
Thus, the conclusion of G's Incompleteness Th is twofold :
For 3. :
By Gödel's Completeness Theorem, there are formal systems in which G is false. How can this not cause a contradiction ?
NO; by G's Completeness Th there are models of $F$ in which $G_F$ is false.
G's Completeness Th, prove that a formula provable in a theory $T$ must be true in all models of $T$.
Thus assuming that $\mathbb N$ is a model of our theory $F$ containing "a certain amount" of arithmetic, we have that all theorems of $F$ (i.e. formulae provable from $F$'s axioms) must be true in all models of $F$.
But $G_F$ is not provable from $F$'s axioms; thus, it must be not true in some model of $F$.
The proof of G's Incompleteness Th give us the insight that $G_F$ is true in $\mathbb N$; thus, it must be false in some model of $F$ different from $\mathbb N$, i.e. in some non-standard model of arithmetic.
Best Answer
There are three distinct concepts to consider. Take the sentence “some elephants can fly.” The grammatical form of this sentence indicates that it is a proposition since it has a truth value, but to assert it is to claim that there really are elephants that can fly. This sentence can be understood as a judgement, i.e. as saying that “some elephants can fly” is true.
To summarize, an assertion claims the reference of sentence, a statement is a sentence with a truth value, and a statement can be judged as true/false.
These concepts have been discussed by the likes of Frege, Russell/Whitehead, Wittgenstein, Kripke, Tarski, and so on. Russell and Frege are probably the most pertinent source to look into for specific references, and the SEP article “Assertion” can give an overview of the full context.