Why we can analytically define $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this?
Algebraically one cannot distinguish between $\varepsilon$ and $-\varepsilon$. But it is possible to augment their definition analytically so to distinguish them.
In dual numbers there is a common equality: for differentiable at $x=a$ function $f(x)$, $f(a+b\varepsilon)=f(a)+b\varepsilon f'(a)$.
Now, one can define that if at point $x=a$ $f(x)$ has right and left derivatives $f'_r(a)$ and $f'_l(a)$, and they are not equal, then
$f(a+\varepsilon)=f(a)+\varepsilon f'_r(a)$ and $f(a-\varepsilon)=f(a)-\varepsilon f'_l(a)$. In other words, $\varepsilon$ is defined as a positive infinitesimal, and $-\varepsilon$ is defined as a negative infinitesimal. This provides an optional analytic definition distinguishing $\varepsilon$ from $-\varepsilon$, but the algebraic structure can work just well without such additional analytic property (but with it one can evaluate more functions at more dual numbers).
Still, in dual numbers one cannot distinguish $\varepsilon$ from $a \varepsilon$ when $a>0$ even with this analytic addition.
So, my question is: why is it possible to define dual numbers in such a way so to distinguish the sign of dual unity, but the same cannot (?) be done with sign of imaginary and hyperbolic unities and with scale in dual numbers?
Is there any strong argument, why?
Is it because the lexicographical ordering in dual numbers naturally embeds into ordering of reals, while in complex and split-complex numbers it does not?
 A: The short answer is "because you are considering $\mathbb{C}$ and $\mathbb{R}[\epsilon]$ them with different structure", which is an artificial choice.
Maybe to illustrate the point : If I want to work with $\mathbb{Q}[\sqrt{2}]$ would you consider that I can tell $\sqrt{2}$ and $-\sqrt{2}$ apart ? I'm not sure you can come up with an an argument for one answer over the other everybody would agree on.
Whether you can distinguish or not these depends on the type of structure you include. As you pointed out, "algebraically" - that is only using the ring structure - you can't distinguish $\epsilon$ and $-\epsilon$, to distinguish them you added new structure, for example the lexicographical order on $\mathbb{R}[\epsilon]$, or the way it acts differentiable functions can be evaluated on dual numbers.
Similarly, If you only consider $\mathbb{C}$ as a field, then you can't distinguish between $i$ and $-i$ because there is a field automorphism exchanging them, but you could add structure on $\mathbb{C}$ that would allow to distinguish them. For example, one could completely arbitrarily consider $\mathbb{C}$ as a field with a marked element "i", or put a "lexicographical order" on it, so that $i> -i$; that would make $i$ and $-i$ distinguished. Of course this structure is not very interesting, so you don't want to do this, and most people will not accept this as an answer.
But what about the structure of "being oriented" as a 2-dimensional real vector space (in addition to its field structure) ? That's a structure on the complex numbers that I would consider relevant, especially if you are using complex numbers to do geometry, and that allows to distinguish  between $i$ and $-i$.
And on the contrary, if you consider $\mathbb{C}$ with less structure, for example as just a real vector space, then you can't tell $i$ and $-i$ apart but, you can't tell $i$ and $2i$ or $2i+3$ apart either.
For the question of $\mathbb{Q}[\sqrt{2}]$ the answer could depend on whether you are considering it as a field or as an ordered field.
So at the end of the day, the only question is whether you would consider that a certain structure that could be used to distinguish two (or more) elements is relevant or not. And that's a purely "sociological" question that has nothing to do with the mathematics, only with what we chose to consider relevant or not.
