Do we have false statement in inconsistent systems? This is a rather soft question.
My understanding:
Suppose we have $x \in ℝ$ and $x^2 = -1$ [in the normal interpretation].
Then the statement "there exists $r \in ℝ$ such that $r^2 = -1$" is true.
This is because $x \in ℝ$ and $x^2 = -1$ form a contradiction, and under contradictory settings any statement follows. That is, in an inconsistent system, any statement is true. [UPDATE: this should be "any statement can be proved" as pointed out in the following answers/comments]
My question:
So, is that still valid to say that the negation of a true statement in an inconsistent system is false? If yes, then we would have any statement in an inconsistent system is simultaneously true and false. [UPDATE: this implication is actually wrong and has been corrected in the following answers/comments]
Or do we rather leave false to be undefined in inconsistent systems? (since I think the definition of false is to some extent redundant in such systems)
Motivation:
I am thinking about what does it actually mean when we say some statement is true.
In a vacuous implication, we say that the premise is false. However, for example, when we are using proof by contradiction to test if a statement is false, we actually treat the statement as if it is a true statement until we hit a contradiction, and then conclude that the statement is false, under the given settings. In other words, a statement is not necessary to be false if we don’t expect consistent system in the first place.
 A: There are different (equivalent) definitions of consistency.
Basically, an inconsistent system is a system that proves a sentence $\varphi$ and its negation $¬ \varphi$.
If so, due to the fact that the negation of a True sentence is False, and vice versa, an inconsistent system is a system that proves True sentences as well as False ones.
Thus,

YES, we have false statement in inconsistent systems.

Regarding your example, we assume that we know facts about real numbers (i.e. mathematical objects whose collection is named with $\mathbb R$), where for simplicity I'll equate a "mathematical fact" with the content expressed by a mathematical theorem.
It is a theorem that, for every real number $r : r^2 \ge 0$.
This means that if we can prove that, for some real $x$, we have $x^2=-1$, this fact contradicts the above theorem.
This amounts to having found an inconsistency in the system we have used to prove it.

Does it mean that every statement in an inconsistent system is simultaneously true and false?

If we agree that there are mathematical objects called (real) numbers and there are objective facts regarding them that we can "discover" through proofs in a suitable system describing them, we accept the "classical" concept of Truth and thus we cannot have statements that are both True and False.
Thus, if we have an inconsistent theory of real numbers, i.e. a system that proves both a statement $\varphi$ and its negation $\lnot \varphi$, we have to conclude that the system is a wrong description of the reals and we have to fix it (as happened already in the past).

References:

*

*Jan Wolenski, Semantics and Truth (2019, Springer)


*Stewart Shapiro, We hold these truths to be self-evident: But what do we mean by that? (RevSL,2009)
A: First off, "There exist $x \in \mathbb{R}$ such that $x^2 = -1$" is by itself not at all contradictory. It is just not true in the actual world by our usual understanding of the symbols $\mathbb{R}, -x$ etc, in which there just so happen to be no negative squares. A contradiction only arises if the theory additionally proves that there are no negative squares, in which case the theory proves both the statement and its negation. This is what I will assume in the following.
Remember that statements aren't just true or false by themselves: Truth is defined relative to interpretations. So what exactly is it that you're asking? In which structures would you like the statements to be false?

Are there any theorems that are false in all models of the inconsistent theory?

In a consistent theory, the answer would be "no, trivially", because the models of a theory are defined as those structures in which all theorems hold, i.e. in which no statement of the theory is false.
But an inconsistent theory has no model: There is no structure in which a contradiction is true. So the answer to this question is: Yes, vacuously, because there are no models to begin with, so in particular there are none in which there are not any statements of the theory that are false in it.
Instead, we may ask:

Are there any theorems that are false in any conceivable structure whatsoever?

In classical logic, with the principle of explosion, an inconsistent theory proves everything. This means in particular that it proves $\phi$ and $\neg \phi$ for any statement $\phi$. But although both may be provable, $\phi$ and $\neg \phi$ can never be simultaneously true under a given interpretation. So in any conceivable structure, for all the infinitely many sentences $\phi$, either $\phi$ is true but $\neg \phi$ false in that structure or vice versa, whereas both of them are theorems. So here the answer is: Yes, there are infinitely many such structures in which infinitely many statements of the theory are false.
In the context of theories, truth is often understood as truth in the standard model with the "intended interpretation" for the non-logical symbols: By saying "$s(0) + s(0) = s(s(0))$ is true" we mean that it is true in the structure of the natural numbers with the successor function and addition defined as usual.
But again: Since an inconsistent theory doesn't have any models, it doesn't have a standard model either. So the question

Are there any theorems that are false in the standard model of the inconsistent theory?

can not be answered.
But the idea of a standard model is that it is a formalization of the real world. So we may ask:

Are there any theorems that are false in the real world?

Again, for every of the infinitely many provable pairs of statements $\phi, \neg \phi$, one of them must be false under each interpretation, such as the real world. So the answer here is again yes: An inconsistent theory proves statements that are false in the real world, namely those whose negation is true in the real world.
This is a crucial point to understand in symbolic logic: Truth exists only relative to interpretations, and the real world/standard model with the intended meaning of the symbols is just one of them. We very well can also have non-standard interpretations in which we, say, take the symbol "$\_^2$" to mean "square root", that yield different truth values for the same sentences. When asking about truth, you have to specify which interpretation you are talking about.
In any given interpretation, any given statement takes exactly one of the truth values "true" or "false". An inconsistent theory is inconsistent precisely because it has no models, i.e. no structure that makes all statements of the theory come true: There can be no possible interpretation in which a statement is both true and false.
