What is the reason (necessity) behind using tautologies in propositional logic proofs and inference? What is the reason (necessity) behind using tautologies in propositional logic proofs and inference?
And why do we only talk about $\rightarrow$ connective when talking about inference or proofs? In fact there are 16 of them including $(and, or, xor, and, T, ...)$.
 A: I believe that you are confusing tautologies with deductively valid inference rules.  An inference rule like Modus Ponens says that $\psi$ can be inferred from $\varphi$ and $\varphi \rightarrow \psi$.  It can be represented as $\varphi, \varphi \rightarrow \psi \vdash \psi$, or maybe as $\varphi, \varphi \rightarrow \psi \Rightarrow \psi$, but it is not a tautology. There is a tautology associated with it though, which would be $(\varphi \land (\varphi \rightarrow \psi)) \rightarrow \psi$. Indeed, with every inference rule we can associate some conditional.... maybe this is what you are looking at?
A: There's a lot of confusion in your question. One of the core issue, though, is that you have a very narrow view of what logic is because you likely haven't been taught any other perspectives, and you're assuming that that narrow perspective covers the entire field.
In particular, I suspect you've only seen Hilbert style proof systems for classical logic (and possibly only classical propositional logic). A Hilbert-style proof system for a propositional logic consists of one rule of inference, modus ponens, and a variety of axioms primarily in terms of the connective $\to$ because that's what modus ponens talks about. This approach is fairly minimalistic (which makes it nice when you want to prove (meta-)theorems about the system), but is fairly unstructured providing little guidance on what a "good" collection of axioms would be. (Obviously, this isn't a problem when you are just handed the axioms.)
There are other approaches to proof systems, most notably natural deduction and the sequent calculus. These approaches tend to have many rules of inference and few, if any, axioms. In these systems, there's nothing particularly special about $\to$. Indeed, if you wanted to, you could take the rules of inference from these systems and build your own "Hilbert-like" system based on $\{\neg,\land\}$ instead of $\to$ (or really, it's closer to $\{\neg,\to\}$ or $\{\bot,\to\}$ for the "usual" Hilbert-style system). One of the key things Gentzen accomplished with natural deduction and the sequent calculus is making the logical connectives modular. That is, the rules of inference in these systems characterizing $\land$, say, don't depend on the other logical connectives or, very much, on the properties of the resulting logic.
This leads into the fact that there are (many) other logics besides classical propositional logic. Logical connectives are not Boolean functions, and there's an unlimited number of them. It is a meta-theorem that you can relate propositional functions to Boolean functions in classical propositional logic (the soundness and completeness theorems). The logic and its models are different things. Indeed, if we slightly tweak classical propositional logic to intuitionistic propositional logic, it is no longer the case that Boolean functions are complete models for it. Similarly, if we add logical connectives like quantifiers or modalities, Boolean functions are no longer even a model of these system, let alone a sound and complete one. It pays to keep the distinction between syntax and semantics clear in your mind, and to understand that there isn't just One True Logic. There are many, many different, often incomparable, logics.
It should be clear at this point that 1) the focus of Hilbert-style systems on $\to$ is ultimately arbitrary, there are reasons one would make this choice as opposed to others, as DanielV's comment alludes to, but other choices can be made; and 2) the existence of certain Boolean functions is not that relevant to whether something is, or should be, a logical connective especially once we go beyond classical logics.
As for tautologies, we don't use them in proofs and inference; we use (only) axioms and rules of inference for proofs and inference. Tautologies are what proofs prove. From a formal perspective, "tautology" and "theorem" are more or less synonymous. When you use a tautology in an informal proof, you can either view it as you are working in a system where that tautology is an axiom/rule of inference, or you can view it as simply a reference to a proof done before, i.e. if you were to formalize the proof you would need to inline the proofs of any tautologies you used with the resulting proof consisting only of axioms and rules of inference. Some systems, like the sequent calculus, explicitly have rules of inference for reusing previously proved theorems: the cut rule. Modus ponens itself is closely related to a special case of the cut rule, and also captures to some extent this idea of reusing previously proved results, which is another reason it was chosen as the primary rule of Hilbert-style proof systems.
A: An answer at the basic level. 
Suppose that, from premise (~A --> B) and ~B , you want to derive : A OR (~B <-->C) . 
You will you OR-introduction rule: "from X, infer X OR Y". 
This rule belongs to syntax: it tells you what you can write after what, which concatenation of symbols is correct. 
~A --> B 
~ B 
~ ~ A ( by modus tollens) 
A  ( by double negation) 
A OR ( ~B <--> C) 
Now, you may ask yourself the question : what guarantees this OR introduction rule? 
By asking this question, you refer to the semantic notion of validity : you are wondering what makes sure that  the proposition A OR ( ~B <--> C)  can never be false when A is true. 
But this last question is the same as : 


*

*what guarantees that "A OR ( ~B <--> C)" is always true when A is true? 

*what guaranties that, in all possible cases : IF A is true, THEN A OR ( ~B <--> C) is also true? 
And at this point you will appreciate this semantic notion of " tautology". 
You can safely infer A OR ( ~B <--> C) from A, because, generally speaking, 
X --> (X OR Y) 
is a tautology, that is, a formula that is true in all possible cases ( truth value assignments). 
