The main algorithmic problems in logic (in my bachelors CS course) are: $0)$ evaluation problem "Auswertungsproblem", $1)$ equivalence problem, $2)$ logical consequence, $3)$ satisfiability problem and $4)$ validity problem. All of these problems can be reduced to the satisfiability problem or the evaluation problem.
The equivalence problem states: "Given $\varphi, \psi \in AL \ \text{is} \ \varphi \equiv \psi$?". I can reduce this problem to the logical consequence problem because if $\varphi \equiv \psi \Longleftrightarrow \varphi \vDash \psi \land \psi \vDash \varphi$. If we can determine such a consequence then $\varphi$ and $\psi$ are equivalent.
The logical consequence problem states that: "$\text{Given} \ \Phi \subseteq AL \ \text{and } \ \psi \in AL, \ \text{is} \ \Phi \vDash \psi \ \text{valid}$?" We can reduce this problem to the satisfiability problem because we know that $\Phi \vDash \psi \Longleftrightarrow \Phi \cup \{\lnot \psi\} \ \text{is not satisfiable}$.
I'm having a problem with the very last section. Why is "$\Phi \vDash \psi \Longleftrightarrow \Phi \cup \{\lnot \psi\} \ \text{is not satisfiable}$" true? I can't seem to understand how to go about showing this.
I mean the idea for the whole exercise is to understand that instead of solving $5$ different problems, it is more efficient to reduce them to ones that're "easier" to solve.
Regarding the title question:
(i) if $\Phi \vDash \psi$, then $\Phi \cup \{ ¬ \psi \}$ is unsatisfiable.
Assume not, i.e. $\Phi \cup \{ ¬ \psi \}$ is satisfiable; this means that there is some valuation such that all of $\Phi$'s are TRUE and also $\lnot \psi$ is, i.e. $\psi$ is FALSE, contradicting the fact that $\Phi \vDash \psi$ means: in every valuation where all of $\Phi$'s are TRUE, $\psi$ must be TRUE also.
(ii) if $\Phi \cup \{ ¬ \psi \}$ is unsatisfiable, then $\Phi \vDash \psi$.
The proof is similar.
The body of the question is not very clear, but yes, the various concept are strictly linked.
With the previous result we have showed how to reduce the problem of logical consequence [locical implication] to that of satisfiability.
Also validity can be so reduced: a formula $\varphi$ is valid iff $\lnot \varphi$ is unsatisfiable.
And also equivalence is: two formulas are logically equivalent iff they imply each other.