Recognitions:

 Quote by Altabeh Why not?
Well, because for a long time you were denying my claim that if we looked at a geodesic in Minkowski coordinates, and then used the coordinate transformation to define the same path in Rindler coordinates, then the path would still be a geodesic relative to the Rindler metric. But I guess you were just misunderstanding what I meant when I said things like "same path", not realizing I was talking about using the coordinate transformation to find the new coordinates of the same physical events along a given path which was originally defined in Minkowski coordinates.
 Quote by Altabeh But let's speak more generally and say that the worldline is not just timelike but arbitrary so the proper "quantity" between any two events lying on it is an extremum.
Yeah, spacelike and null geodesics in Minkowski coordinates should also map to spacelike and null geodesics in Rindler coordinates, assuming again that we are looking at the same physical set of events in each coordinate system.

Assuming we've settled this issue, perhaps we can revisit my earlier claim that the equivalence principle would hold in arbitrarily large regions of Rindler coordinates? Hopefully you'd agree that in any flat SR spacetime described in Minkowski coordinates, it should be possible to construct a physical network of rulers and synchronized clocks moving inertially relative to this coordinate system (a network of the type that Einstein used to define the concept of an 'inertial frame'), such that if we use the readings on the rulers and clocks to define a new coordinate system, the laws of physics as seen in this coordinate system (which is just a different Minkowski frame) will be those seen in any SR inertial frame. Now, if you agree that Minkowski coordinates (on the Rindler wedge) and Rindler coordinates are just different descriptions of the same physical spacetime, this same physical network of rulers and clocks could also be described in Rindler coordinates, agreed?

Now suppose we have such a physical network, and we want to describe where events in an experiment happen relative to that network in the context of our Minkowski or Rindler coordinate systems. Suppose a particular event E (a collision, say) happens next to the x=12 light-seconds mark on the physical ruler representing the x-axis in this network, and that the physical clock at that mark reads t=8 seconds when E happens next to it. Then if we are using a separate Minkowski coordinate system with coordinates x',t', to describe events in this spacetime, the coordinates of the physical clock at the x=12 l.s. mark on this ruler reading t=8 s may happen at some completely different coordinates in this system, say x'=23 l.s. and t'=100 s. Likewise, in the Rindler coordinate system, the coordinates x'' and t'' of this event will be different as well. However, both the Minkowski coordinate system and the Rindler coordinate system will agree that whatever the coordinates x' and t' (or x'' and t'') of the event of the physical clock at the x=12 l.s. mark reading t=8 s, the same coordinates x' and t' (or x'' and t'') would be assigned to the event E of the collision--the Minkowski coordinate system and the Rindler coordinate system cannot disagree about local facts like whether two events (in this case the event of that clock at that marking reading 8 s and the event E of the collision) coincide at the same point in spacetime or not!

If both the Minkowski coordinate system and the Rindler coordinate system (along with the metric and laws of physics expressed in those coordinate systems) agree in their predictions about what ruler-markings and clock-readings on the physical network coincide with which events in any physical experiment that takes place within the region covered by that physical network, then both coordinate systems should agree in their predictions about what the equations of the laws of physics will be when expressed in the coordinate system defined by the network (not when expressed in terms of their own coordinates). And of course in flat SR spacetime it should always be possible to construct a physical network of rulers and clocks which define a coordinate system where the laws of physics work the same way as in any other inertial coordinate system, right?

This idea is essentially no different than the idea of the equivalence principle, that even if you have a curved spacetime described in some coordinate system like Schwarzschild coordinates, in any local region you should always be able to construct a network of freefalling rulers and clocks such that that the laws of physics as expressed in the coordinates defined by that network (not when expressed in Schwarzschild coordinates in that region) reduce to the same laws seen in an inertial frame in SR (at least to the first order or something). The only difference is that in this case we need not confine the grid of rulers and clocks to a small local region, they can cover any arbitrarily large region of the Rindler wedge and it'll still be true that the laws of physics as expressed in the coordinates defined by the network (not when expressed in Rindler coordinates in that region) will be exactly those seen in any SR inertial frame.

 Quote by JesseM Well, because for a long time you were denying my claim that if we looked at a geodesic in Minkowski coordinates, and then used the coordinate transformation to define the same path in Rindler coordinates, then the path would still be a geodesic relative to the Rindler metric. But I guess you were just misunderstanding what I meant when I said things like "same path", not realizing I was talking about using the coordinate transformation to find the new coordinates of the same physical events along a given path which was originally defined in Minkowski coordinates.
Well, as DrGreg said, the problem lies in the fact that some authors describe the 3-vector $dx^i/d\tau$ as proper velocity and all my focus on the way I'm expected to treat the problems here automatically had been going to this point and don't get me wrong this was the way I learned and tought people until recently that I changed my mind and started using the magnitude of the 4-acceleration instead of $$d^2x^i/d\tau^2$$ as the "proper" acceleration! If you noticed in my early posts here, I doubted that the gravitational field of Rindler spacetime is uniform just because the proper 3-accceleration I mentioned above is dependent on $$x$$, thus it doesn't allow the spacetime to be accompanied by a uniform field. However, the challenge this caused in mind isn't still settled and I hope further studies in the future will help me ponder the problems more than I do now!

I'm in a hurry to go somewhere, so I don't have enough time to read the whole post. I'll be back soon.

AB

My own view on the EP not only on Rindler wedge but in any spacetime with a vanishing Riemann tensor but non-vanishing Christoffel symbols is briefly stated in Papaetrou's book "Lectures on General Relativity" at page 56:

 When there are gravitational accelerations present, as for example in the gravitational field of the earth, the space cannot be the flat Minkowski space. Indeed, in the Minkowski space we can have $$\Gamma^{\alpha}_{\mu\nu}=0,$$ everywhere. This should then be interpreted as meaning that the sum of the inertial and the gravitational acceleration could be made equal to zero everywhere. This does, however, not correspond to our experience about gravitational accelerations: When gravitational accelerations exist, it is not possible to make them vanish everywhere. We can only make them vanish at one point, or approximately in a small region, by the use of an appropriate coordinate system. Therefore, when a gravitational field is present, the space will be necessarily a curved Riemannian space.
But since the spacetime is flat and yet there we have non-vanishing Christoffel symbols, there must be a coordinate transformation that does make the these symbols vanish and this is what happens to be true on the Rindler wedge when events can finid co-pairs in the Minkowski spacetime everywhere, leading to the fact that on Rindler wedge the EP is valid overally!

AB