Given that the domain of $n$, as stated, is all* $n\in \mathbb{Z}$, then your reasoning is correct and $d$ is indeed false. Negative integers would serve as your counterexample showing the statement is false. So the answer key must be wrong, or there was a typo in the problem set!
If the domain of $n$ were $\mathbb{N}$, and depending on how one defines the natural numbers $\mathbb{N}$: would is any integer $n \geq 0$ (or an integer $n\geq 1$).
Hence, in either case, negative numbers are excluded from the domain of $n\in \mathbb{N}$.
Hence, $(d)$ would be true, if the domain were in fact $n \geq 0$: given ANY $n\in \mathbb{N},\;3n\leq 4n$, since $3\leq 4$ is clearly true.
(1) The OP writes:
Intuitively, "The present King of France is bald." is false. But Bertrand Russell said it would mean that "The present King of France is not bald.", which seems to be false. This apparently leads to a contradiction.
No Bertrand Russell didn't say quite that. Rather he distinguished two readings of "The present King of France is not bald." This can be parsed as either "It is not the case that the-present-King-of-France-is-bald" or "The present King of France is not-bald". (There's a scope ambiguity -- does the negation take wide scope, the whole sentence, or narrow scope, the predicate?)
Russell regiments "The present King of France is bald" as
$$\exists x(KFx \land \forall y(KFy \to y = x) \land Bx)$$
where '$KF$...' expresses '... is a present King of France' and '$B$...' expresses is bald (there is one and only one King of France and he is bald). Then the two readings of "The present King of France is not bald" are respectively
$$\neg\exists x(KFx \land \forall y(KFy \to y = x) \land Bx)$$
$$\exists x(KFx \land \forall y(KFy \to y = x) \land \neg Bx)$$
The first is true, the second false -- no paradox or contradiction. Trouble only arises if you muddle the two.
(2) The OP also writes
Does $\frac{1}{0}=3$ mean anything, since $\frac{1}{0}$ doesn't exist?
Compare: "The (present) King of France" is a meaningful expression -- you know perfectly well what condition someone would have to satisfy to be its denotation. In fact, it is because you understand the expression (grasp its meaning) that -- putting that together with your knowledge of France's current constitutional arrangements -- you know it lacks a referent. The expression is linguistically meaningful but happens to denote nothing (with the world as it is). Similarly there's a good sense in which do you understand "$\frac{1}{0}$" perfectly well: it means "the result of dividing one by zero". It is because you understand the notation, and because you know that division is a partial function and returns no value when the second argument is zero, that you know that "$\frac{1}{0}=3$" isn't true. The symbols aren't mere garbage -- you know what function you are supposed to be applying to which arguments. So, in a good sense, the symbols "$\frac{1}{0}$" are meaningful even though they fail to denote a value. In Frege's terms, the expression has sense but lacks a reference.
(3) Marc van Leeuwen writes
Using the definite article "the" in "the present King of France" implicitly claims there is exactly one person presently King of France; since that is not the case, any phrase that refers to this is meaningless."
Not so. For example, the sentence "No one is the present King of France" is not only meaningful but true -- so it can't be that just containing the non-referring "the present King of France" makes for meaninglessness.
Best Answer
Insofar as this is not just a question about linguistics, the way this sort of issue is handled in mathematics is that to even use the word "the" to refer to a mathematical object is to make an implicit claim that that object both exists and is unique, e.g. when we speak of "the cyclic group of order $n$" and so forth. So there is an implicit existential quantifier there (one might render it more explicitly as "there exists a unique person who is king of France, and..."), and if the domain of discourse over which that quantifier ranges is empty (in this case, if France does not have a king) then the statement is false.
One needs to be a bit careful about this in mathematics, though, because sometimes the implicit quantifier is a universal quantifier and then if the domain of discourse being quantified over is empty such statements are vacuously true.