Confusion with the equivalence principle and inertial frames 
This is an explanation of the equivalence principle. A. says accelerating frames can simulate gravity. Then, I am confused. Are the frames presented in A. inertial frames? Compared to B. I have the impression that the frames in A. are not inertial frames. Then, does it mean that any observer standing on earth is not an inertial observer? But this is contrary to my common sense... Also, the Einstein's postulate says that the speed of light is same in all inertial frames. Then, the speed of light measured from frames in A are not equal to $c$?
 A: An inertial reference frame is a reference frame where an object not subjected to a force is at rest or moves with uniform velocity. The Einstein equivalence principle in its basic formulation states that the gravitational force as experienced locally is not distinguishable from the force experienced in a non-inertial (accelerated) reference frame.
Answers to the questions:
The frames in A. are non-inertial.
An observer standing on the earth is not an inertial reference frame; in fact if you drop a body it starts to accelerate.
Locally the speed of light is $c$. Locally means measured by an inertial reference frame instantaneously at rest with the accelerated frame.
A: 
Are the frames presented in A. inertial frames?

No, the frames are accelerated reference frames - an accelerometer at rest in either frame reads non-zero acceleration - and not inertial reference frames.

Then, does it mean that any observer standing on earth is not an
  inertial observer?

Yes, an accelerometer attached to an observer at rest on the surface of the Earth reads non-zero acceleration and so the observer's reference frame is an accelerated reference frame and not an inertial reference frame.

Then, the speed of light measured from frames in A are not equal to c?

You might find the answers here useful:  Does the speed of light vary in non-inertial frames?
A: "If we accept the principle of equivalence, we must also accept that light falls in a gravitational field with the same acceleration as material bodies." 
 http://sethi.lamar.edu/bahrim-cristian/Courses/PHYS4480/4480-PROBLEMS/optics-gravit-lens_PPT.pdf
This (Newtonian) acceleration was confirmed by the Pound-Rebka experiment - see the 4th paragraph here: 
 http://virgo.lal.in2p3.fr/NPAC/relativite_fichiers/pound.pdf
The Newtonian acceleration is incompatible with gravitational time dilation. This means that the Pound-Rebka experiment disproved general relativity.
