On (1). In more standard terminology:
A formal theory is [negation] complete if for each sentence (closed formula) φ either φ or ¬φ is provable.
A logical deductive system is semantically complete if every sentence (closed formula) φ of the system which is a logical truth is provable.
Gödel's incompleteness theorem shows that first-order Peano Arithmetic isn't negation complete. (And trivially, since one of either φ or ¬φ is true on the standard interpretation, there is a truth PA can't prove).
But the first-order deductive system built into PA is still semantically complete (by Gödel's completeness theorem!).
It was/is important that Gödel's official incompleteness result is a purely syntactic one, proved on purely syntactic assumptions. For recall the context back around 1930. Semantic notions (pre-Tarski) were widely thought murky and "unscientific"; and the major research programme in foundations in Gödel's environment was the Hilbertian program which presupposes that we can completely axiomatize swathes of mathematics. Showing on syntactic assumptions that the Hilbertians would have to buy that we can't even get a complete theory of elementary arithmetic was therefore devastating.
About intuition.
Aristotle introduced a notion of consequence along the following lines. Suppose $A$ and $B$ are sentences, each of them either true or false. We study the relation :
(1) 'If A is true, then B has to be true too'.
Many logicians reckoned that the main use of logic is derive conclusions from premises, with the obvious goal that if we start from true premises, and we use “logical rules”, we will derive true conclusions. This idea of logic gives rise to the relation :
(2) 'From A, we can prove B'.
Now we try to translate all this into modern terms.
(1) translates into the relation of logical consequence $A \vDash B$ :
'For every interpretation $\mathcal I$, if A is true under $\mathcal I$, then B is true under $\mathcal I$.'
We “generalize” it putting in place of the sentence A a set $\Gamma$ of sentences :
$\Gamma \vDash \varphi$
which means that :
'For every interpretation $\mathcal I$, if all of $\Gamma$ are true under $\mathcal I$, then $\varphi$ is true under $\mathcal I$.'
Lastly, we have the “special acse” when $\Gamma = \emptyset$ : $\vDash \varphi$ means that the sentence $\varphi$ is valid (in propositional logic it is a tautology) when it is true under every interpretation.
In this case, we call it also logical (or universally) valid sentence, meaning that it is true “under all possible circumstances”.
For (2) [see Neil Tennant, Natural logic (1978), page 5], we have that, in order to demonstrate the validity of an argument one needs a proof.
A proof of an argument is a sequence of steps starting from its premisses, taken as starting points, to its conclusion as end. Within a proof each step, or 'inferences', must be obviously valid.
A system of proof is a codification of these obviously valid kinds of inference, which we call rules of inference. A proof in accordance with these rules must, in order to meet the demands of certainty, satisfy the following conditions :
(i) It must be of finite size.
(ii) Every one of its steps must be effectively checkable.
A system of this type is a logical calculus (i.e.proof systems) formalizing the relation of derivability :
$A \vdash B$
which is :
'B is derivable using only the logical rules of the calculus, starting from A as premise’.
We “generalize” it putting in place of the sentence A a set $\Gamma$ of sentences :
$\Gamma \vdash \varphi$.
Lastly, we have the “special case” when $\Gamma = \emptyset$ : $\vdash \varphi$ means that the sentence $\varphi$ is theorem of the calculus (it is provable) when it is derivable from no premises.
Obviously there must be no invalid proofs in our system : the system must be sound.
Moreover we wish to be able to prove any valid argument expressible in the language. The system, that is, must be complete (or adequate).
Soundness : if $\Gamma \vdash \varphi$, then $\Gamma \vDash \varphi$ (with the particular case : if $\vdash \varphi$, then $\vDash \varphi$)
Completeness : if $\Gamma \vDash \varphi$, then $\Gamma \vdash \varphi$ (with the particular case : if $\vDash \varphi$, then $\vdash \varphi$).
Best Answer
First, some terminological issues. $A\vdash B$ usually means $A$ is provable or derivable from $B$. This is a purely syntactic property that is about building formal proofs and does not require knowing whether anything is "true" or not. Validity usually means that a formula is semantically true in all models and is written $\vDash B$ with $A\vDash B$ as shorthand for "$\mathfrak M\vDash A$ implies $\mathfrak M\vDash B$" for all models $\mathfrak M$ with $\mathfrak M\vDash A$ meaning "$A$ is semantically true in model $\mathfrak M$". "Syllogism" has a fairly specific meaning and is relatively archaic at this point. You'll rarely find it used in a modern logic textbook except in a "history of logic" section. You are also using "sound" in the more philosophical sense. This unfortunately conflicts with "sound" in the mathematical logic sense which becomes relevant... now. $\vdash$ and $\vDash$ are (for a given logic) usually sound and complete. Soundness means "$\vdash B$ implies $\vDash B$", i.e. what we can prove is valid. Completeness means "$\vDash B$ implies $\vdash B$", i.e. we can prove everything that is valid. Soundness and completeness together mean that $\vdash$ and $\vDash$ are the same relation on formulas which is why the terminology often gets muddled. However, soundness and completeness are non-trivial (meta-)theorems (particularly completeness), and you need to understand what $\vdash$ and $\vDash$ mean on their own before you can prove them.
To actually start addressing your question, it doesn't make sense in mathematical logic to talk about a formula just being "true". You can talk about it being provable (i.e. a theorem) or being valid. Validity, as I mentioned before, is defined in terms of a notion of semantic truth, and the key thing here is that truth is with respect to a model written $\mathfrak M\vDash B$ which means $B$ is true in the model $\mathfrak M$. Validity can then be written as "for all models $\mathfrak M$, $\mathfrak M\vDash B$". For propositional logic, the models are often called "valuations" or "(truth) assignments" as in Mauro ALLEGRANZA's answer. In this case, they consist entirely of assignments of truth values to atomic propositions which can then be lifted to assignments of truth values to all formulas via the interpretation of the connectives.
The closest thing to what you want is therefore something like $\mathfrak M\vDash B$ for some particular model $\mathfrak M$.
There is nothing in mathematical logic to say that some formula is "true in reality". Whether something is "true in reality" is not a mathematical question but a physical or maybe a philosophical one. Even semantics in mathematical logic interprets things into mathematical structures, typically sets, so semantic truth is just a statement about certain mathematical structures.
If a mathematical logician wanted to say something about a formula being "true in reality" (which would be a very odd thing for them to do), they'd just say it in natural language.