Well, you "step into [the stone object]". You're stepping into it. So I think your description of "move through opened door" is apt. Beyond that, it's not defined. The DM can narrate.
Whether you can see inside the stone is not defined, but it seems entirely reasonable that you cannot. It would have been clearer if the spell had said you were blind while within the rock, but it did not. However, it seems reasonable you can't see, since there's rock in front of your eyes. If I mean, I guess there's rock in front of your eyes. You're magically "melded" into the rock, and that is not fully defined.
So, whether you can see to read a scroll is up to the DM, however, it's likely and reasonable that you can't see. If you can't see, you can't read a scroll.