is any formula the logical consequence of an inconsistent set [of formulas] ?
YES
The definition of logical consequence is :
Let $\Gamma$ be a set of formulas and $\varphi$ a formula. Then $\Gamma$ logically implies $\varphi$ (and $\varphi$ is a logical consequence of $\Gamma$), written $\Gamma \vDash \varphi$, iff every interpretation that satisfies every member of $\Gamma$ also satisfies $\varphi$.
Thus, the "logical form" of the definition is :
for all interpretation $I$, if $\text { Satisfy }(\Gamma, I)$, then $\text { Satisfy }(\varphi, I)$.
If $\Gamma$ is inconsistent, then $\text { Satisfy }(\Gamma, I)$ is false for every $I$; thus, the conditional: if $\text { Satisfy }(\Gamma, I)$, then $\text { Satisfy }(\varphi, I)$, is vacuously true for every $I$.
And this, in turn, holds for a formula $\varphi$ whatever.
How completeness applies to this case ?
If $\Gamma = \{ \phi, \lnot \phi \}$ we can apply the rules of the calculus to derive a formula $\psi$ whatever.
With e.g. Natural Deduction, we have :
1) $\phi$ --- premise
2) $\lnot \phi$ --- premise
3) $\bot$ --- from 1) and 2) by $\lnot$-E
4) $\psi$ --- from 3) by $\bot$-E.
How correctness applies to this case ?
Correctness (or soundness) means that the rules applied to (a set of) true premises produce a true conclusion.
But in an inconsistent set of premises not all formulas are true; thus, the definition does not licence us to assert that in this case the conclusion must be true.
