Proof of Lemma 37.5 in Pak's Lectures on Discrete and Polyhedral Geometry I am staring at the proof of Lemma 37.5 in Lectures on Discrete and Polyhedral Geometry, see page 331.
I cannot understand why the required triangulation exists.
In the first paragraph it says "Vertex $v_4$ lies in one of these regions and connected by $γ_{14}$, $γ_{24}$ and $γ_{34}$ to
the vertices in its boundary."
It seems that the author assumes that these curves $γ_{i4}$ are minimizing geodesics, plus they stay in one region.

Am I right? If yes, can it be true?

 A: Yes you are right, each $\gamma_{ij}$ is a shortest path that (as it is stated at the start of the proof). And yes, it is implicitly assumed the geodesics between $v_1$, $v_2$, $v_3$, and $v_4$ do not cross as this
.
For sure such crossing may happen and the author does not specify the choice of points and geodesics to ensure that it does not.
To fix this argument, I would try to find 4 vertices with 6 geodesics between them that divide the surface into triangles with interior angles $\leqslant \pi$. I do not see why it should exist, but I do not have a counterexample in my pocket.
On the other hand, even if you fix this argument, it will not be simpler than the original proof. By the way, Alexandrov gives two proofs of the connecting lemma (which is essentially Lemma 37.5) one in his book and the other in the previous paper (this one is better).
By the way, this note contains a sketch of Alexandrov's theorem (including the connecting lemma).
It is very short, you may read it, think for couple of days, and after that reading Alexandrov should be very easy.
