How is it justified in euclidean geometry to take an arbitrary point? Is it justified using Euclides' axioms to take an arbitrary point?
I mean, it is done for example in this proposition, but how it is justified inside the axiomatic?
I mean, the only points we can get are those which result from the intersection of lines and circles, so how can we get that point?
 A: Euclid's axioms do not really concern themselves with the extremely unnatural hypotheticals like "what if no points on the other side of the line exist?" They fall down even if it gets to the mildly unnatural hypotheticals like "what if you draw circle $CD$ in the proof of that proposition, but it fails to intersect the line?" For that matter, Euclid's definitions, postulates and common notions never specify what it means, in the proof of I.12, for $D$ to be on the other side of $AB$ from $C$.
Generally, Euclid's axioms are rigorous enough for discussions along the lines of: "Once we accept that we're working in basically normal geometry, who's to say that when you construct this right angle, it really is a right angle?" They are not well-suited for criticisms like: "Well, if we're not working in normal geometry but in some weird space that technically satisfies all the postulates you've given, then such-and-such point simply doesn't exist."
For that, I would go to a more modern approach like Hilbert's axioms for geometry. These in particular provide more ways to summon points into existence.
For example, to justify the construction of point $D$ in the proof of Euclid's proposition I.12, we could pick any point $X$ on line $AB$, and use Hilbert's 2nd order axiom to say that there is a point $D$ on line $CX$ such that $X$ lies between $C$ and $D$.

*

*How do we know that $D$ is on the other side of line $AB$ from $C$? Well, to Hilbert, "$P$ is on the other side of line $\ell$ from $Q$" means "lines $PQ$ and $\ell$ intersect at a point between $P$ and $Q$". Here, we have such an intersection of $AB$ and $CD$: point $X$.

*What about the arbitrary point $X$? This part is just a rhetorical device to say that the location of $X$ doesn't matter. We certainly know that points on line $AB$ exists: for example, we can take $X=A$ or $X=B$.

For that matter, I imagine that in several cases, Euclid's use of "arbitrary point" is a similar rhetorical device. In cases where we have no reason to be skeptical that such a point exists (maybe we already have several such points!) it just means that the point doesn't have to be any specific point, making the construction easier to read.
(E.g. "let $X$ be an arbitrary point on line $AB$" is always possible, but "let $X$ be an arbitrary point on line $AB$ other than $A$ or $B$" needs justification - though in this case, you could argue that Euclid's 2nd postulate justifies it.)
