Is "The present King of France is bald" studied by maths? 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.
Could assertions about things which don't exist not be false in mathematics (or even true)?
For example, does  $\frac{1}{0}=3$ mean anything, since $\frac{1}{0}$ doesn't exist?
 A: Mathematical expressions or phrases that are not defined, or that make an implicit claim that is not satisfied, are meaningless, and no (truth) value is ascribed to them. Examples are for example expressions containing a division by $0$, limits of divergent sequences, or taking the minimum of a set of numbers that turns out to be empty; there are many more examples. 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.
A: This is the sort of thing that free logic can handle in ways classical logic cannot. In logic jargon, the phrase "the present king of France" is a singular term that does not denote any object. Depending on which semantics for free logic you use, the sentence "the present king of France is bald" might be true, or false, or truth-valueless.
Separately, in normal first-order logic, we often translate a "the" sentence of English into a formal sentence with a quantifier. In this case, "the present king of France is bald" might become "for every person $P$, if $P$ is the present king of France then $P$ is bald" (which is a true sentence, classically, in the intended interpretation) or "there is a person $P$ who is the present king of France and is bald" (which is false, classically). 
A: (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.
A: 
Could assertions about things which don't exist not be false in
mathematics (or even true)?

Yes. In your king-of-France example:
Let K and B be unary predicates such that:

*

*K(x) = "x is presently a king France

*B(x) = "x is bald"

We assume that that no one is presently a king of France:
$~~~~\neg \exists x: K(x)$
Therefore, no one is both presently the King of France and bald:
$~~~~\neg \exists x: [K(x) \land \forall y:[K(y)\iff y=x]\land B(x)]$
This is a meaningful   assertion about the non-existence of present kings of France.

Does  $\frac{1}{0}=3$ mean anything?

No, not in the real numbers anyway. To be able to then attach any truth value to this statement, it might be recast as $1=3\times 0$. That, of course, would be a false assertion on the reals.
