Thanks to Andrew Steane and Pulsar in this topic I understood that in a frame with constant proper acceleration, each hyperbola in $T-X$ diagram demonstrates a constant position while each line passing through origin demonstrates constant time.
That is, from viewpoint of an accelerated frame, two simultaneous events in different locations are actually two points in the same line (with constant t) but two different hyperbolas.
What bothers me however is that to my understanding, each of these hyperbolas demonstrates an accelerated frame with different proper acceleration. As Pulsar put it:"Note that each hyperbola represent worldlines of travelers with different constant accelerations".
So what are we doing here? Assume a frame with constant proper acceleration $g_0$. Let say worldline of this frame is $x_0 = \frac{c^2}{g_0} = 0.4$. Now if from his viewpoint two events happen at different locations $x_1 = 0.2$ and $x_2 = 0.6$, he has to use two different hyperbolas $x_1$ and $x_2$.
But it's as if he has assumed two different constant proper accelerations for those points. Even worse, different from his acceleration as well. Let say our accelerated frame resides in a spaceship. Center of mass spaceship is in $x_0$. Two simultaneous events $x_1$ and $x_2$ happen at the different end points of spaceship. If from viewpoint of our observer, these points have different proper accelerations, it would mean spaceship should be torn apart! Because every point of spaceship would have different acceleration. It's as if there is some kind of tidal force here. But why is that? I mean physically speaking. For example in Classical mechanics, an accelerated frame will feel a fictitious force due to the inertia. What actually happens in special relativity that we have a tidal force (if any)?
Two observers with constant velocity can not use each other coordinates, unless they use Lorentz transformations first. Our accelerated frame however, use other worldlines (which corresponds to other observers with different accelerations) without using any kind of transformations. How he can do that? I mean if each of these hyperbolas demonstrates a point with different acceleration, how does it make sense to put all of them in the same diagram and make this grid to begin with? Instead of comparing two different hyperbolas, you can compare two different lines as well. Two different lines corresponds to two different inertial observers with different velocities, and we use these lines without doing any kind of transformations.
Edit: Regarding my first question, I think I am reading this diagram incorrectly. Maybe the observer in spaceship don't see a tidal force, rather it's actually the inertial observer outside of spaceship who sees every point on spaceship has different acceleration? At least it makes more sense from what I know from Lorentz transformation.
Edit 2: I changed title to make it more interesting for people.