If something is neither true nor false, what is it?

These statements taken together are called inconsistent. That means that they cannot all be simultaneously true. But the first and second are neither true nor false without broader context. Using the language of first order logic, they might be said to be formulas with "free variables." Here's an example of a formula with free variables $$ 4x+3y=9$$ This is neither true nor false because I haven't told you what $x$ or $y$ are. If I use quantifiers to get rid of all the free variables, then I have a sentence which may be true or false: $$\forall x\forall y (4x+3y=9)$$ $$\forall x\exists y (4x+3y=9)$$ $$\exists x\exists y (4x+3y=9)$$ The first statement is false, while the second two are true.

In the 3 statements you've shown, statement #3 is definitely false, as you have already mentioned.

To clarify, I really meant if there were some state between true and false that could make these consistent.

No, statements #1 and #2 are inconsistent statements, they contradict each other.

Actually, since there's no discrepancy that statement #3 is false, it is a bit of a red herring, irrelevant. Alternately, statement #3 can be omitted and statement #1 and #2 can be re-written as:

1) Statement 2 is true.
2) Statement 1 is false.

or

The next statement is true.
The previous statement is false.

If you are looking for a word or phrase to describe the relationship of the statements, I would say the statements are "paradoxical".

You can also use the term: non sequitur.

noun: non sequitur; plural noun: non sequiturs; noun: nonsequitur; plural noun: nonsequiturs

a conclusion or statement that does not logically follow from the previous argument or statement.

Too long to repeat the whole Wikipedia article.

What you are looking for are the Liar paradox and its variants. Here are the possible resolution.