How is "If $T \cup \{P\} \vdash Q$, then if $T \vdash P$, then $T \vdash Q$" same as saying "If $T \cup \{P\} \vdash Q$, then $T \nvdash P$ or $T \vdash Q$"? And since we are on this topic - how should one interpret P $\to$ Q (where P & Q are some formulae in a first-order language), in an axiomatic system.

That's just the meaning of a mathematical "if ... then": "If $A$ then $B$" = "(not $A$) or ($B$)". So "If $T \vdash P$ then $T \vdash Q$" = "not $T \vdash P$ or $T \vdash Q$" = "$T \nvdash P$ or $T \vdash Q$".

Doesn't this result come from the "truth-table" based definition of the $\to$ (implication connective). And would it be right to even consider this result in a purely a axiomatic-system?

Rather the other way round: The truth table of $\to$ in logic is derived from the mathematical understanding of "if ... then". And how "if ... then" is understood is just a matter of conventional language use.

I'm not sure what you mean by "interpreting $P \to Q$ in an axiomatic system". "$T \vdash P \to Q$" means that the formula $P \to Q$ is derivable without further assumptions from the axioms of $T$ in the given system.

In classical logic (the one based on truth or falsity of propositions) P $\to$ Q is propositionally-equivalent to $\neg$P $\lor$ Q (it's straight forward if one draws the truth tables, for the two propositions). But then how do you interpret $\neg$P in an axiomatic-system? Isn't it an operator that transforms P into an another formula whose proofs are the refutations of P?

The "interpretation" of a formula in a proof system is always that it is derivable using the rules of the system -- it is a purely syntactic notion, "syntactic" in the sense that it is defined purely in terms of fiddling around with symbols, without making explicit reference to "semantic" notions such as valuations or truth tables.

What a proof of a formula looks like depends on the particular proof system, the formula in question, and what proof is being used (there may be several proofs for the same formula in the same system). A proof of $\neg P$ typically amounts to showing that a contradiction can be derived from assuming $P$ using the axioms of the theory.

But you are thinking too semantically: We do not need to "interpret" formulas in a proof system; all there is to the symbol $\vdash$ is that the formula is derivable using the axioms and inference rules of the system.

I don't understand why is it valid to interchangeably use the concepts of classical-logic (truth value based) and predicate-logic (formal system based). Because (See Pg. 30) classical-logic and predicate-logic are only analogous and not the same.

You are mixing up a number of concepts here. The distinction is not between classical logic on the one hand and predicate logic on the other hand, and the definition of predicate logic is not that it is formal-system based. Classical logic is a logic where the law of the excluded middle holds, as opposed to e.g. intuitionistic logic.

There is classical propositional logic (with propositional variables p, q, r, ... and truth tables), and classical predicate logic (with predicates P(x), Q(y,z), and quantifiers $\forall, \exists$).

The semantics for the connectives in classical prediate logic is the same as those for the connectives in classical propositional logic; the truth table for $\to$ is the same in classical propositional logic and classical predicate logic. Predicate logic is a more powerful extension of propositoinal logic, but with the same basic laws.

Both propositional and predicate logic have a truth-conditional semantics and proof systems. It is not true that one is based only on truth conditions and the other is based only on the notion of proof; both are defineable for both logics.. There are proof systems for propositional logic, and there is a truth-conditional semantics for predicate logic.

The distinction is between truth conditions/interpretations/models (via truth tables or clauses on truth conditions in a structure) on the one hand, and formal systems (which only have axioms and rules of inferences, and do not explicitly talk about interpretations) on the other hand.

I now read through p. 30 and admit it is formulated a bit confusingly. The analogy that is made is between semantics ($\vDash$, truth tables) and syntax ($\vdash$, proof systems). The analogy can be made because $\vDash$ and $\vdash$ do in fact induce the same relations -- the same formulas that are tautological are provable -- though their definitions are different.

But this distinction is a different matter from the one between propositional logic and predicate logic: The book introduces propositional logic from the semantic side and predicate logic from the syntactic side, but as elaborated in the next paragraph on p. 30, one can also define a proof system for propositional logic and semantic clauses for predicate logic.

And you can forget about the role of "classical" for the moment; everything that's done in the book (at least up to this point) is classical logic.

The reason why the semantic and syntactic definitions can be used interchangeably to some degree is because they are equivalent: The formulas such that $\vDash P$ are exactly the same formulas such that $\vdash P$: The syntactic proof system is sound and complete w.r.t. the semantics. (But this can not be taken for granted and has to been shown (complicated).)