Do we have false statement in inconsistent systems?

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.