First Order Logic is complete in the sense that: there is a proof procedure for FOL such that just the statements(/wffs) of FOL that are true and remain true under any re-interpretation.
Yes; completeness is wrt the "underlying logic", in the sense that the logical axioms+rules can formally prove all valid (i.e. true in every interpretation) formulas.
More generally, the Completeness Theorem asserts that: Γ ⊨ φ iff Γ ⊢ φ, where Γ is a set of formulas and φ is a formula, i.e. a formula φ is a logical consequence of a set Γ of formulas iff it is derivable from Γ.
The incompleteness is relative to a mathematical theory (with some specific requirements) that is built on top of the "underlying logic" with the addition of mathematical symbols and axioms, like e.g. Peano arithmetic.
In this case, Gödel’s Incompleteness Theorems asserts that there is a sentence G such that neither G nor its negation ¬G are derivable from the axioms of the theory. We say that G is undecidable (wrt first-order arithmetic).
Where is the link between the two results? Why there is no inconsistency?
Because neither G nor ¬G are valid formulas: they are formulas in the language of arithmetic, and thus they express a "fact" about numbers; being sentences, one of them must be true (and consequently, its negation is false) but it is not a logical consequence of the axioms of arithmetic.
Thus, your point (2) above is correct: the incompleteness of first-order arithmetic (and ZFC as well) means that there are models of the arithmetical axioms in which the said formula is not true. See Non-standard models of arithmetic.
See also Paris-Harrington theorem for a more "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic.
For more details, see How is First Order Logic complete but not decidable?
The result applies to ZFC simply because we can Deduce PA's axioms in ZFC; see also How to prove that Gödel's Incompleteness Theorems apply to ZFC.
Regarding ZFC, we have many "non esoteric" examples; see List of statements independent of ZFC.
The first one was due (partly) to Gödel again: the Continuum hypothesis or CH is idependent of ZFC.
In 1940 Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen in 1963 invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC.
Thus, CH is an example of undecidable statement wrt ZFC (like formula G above). See also Solutions to the Continuum Hypothesis.
Another example of statement independent of ZFC is the so-called Axiom of constructibility: usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.
The axiom, first investigated by Kurt Gödel (again!), is inconsistent with the proposition stronger large cardinal axioms (see list of large cardinal properties). See also Completion of ZFC.
In conclusion, re your question about "non-standard" interpretation of the "is a member of" relation (∊), if we assume that the von Neumann universe V is the standard interpretation of ∊, we have that the Constructible universe L is a non-standard one.
