Does it have to be stone or can it be any form of solid surface?
You step into a stone object or surface
I read that as a stone object or a stone surface. I think the intent is perfectly clear, but a rules lawyer might argue there are two choices, one a stone object, the other a surface. I think even a lenient DM would not grant you that, but you could ask. I think it is very dubious though. I mean, it is called meld into stone, not meld into various things.
What happens if it's gravel? Does it count as multiple small objects or one surface? Same goes for sand which could be eroded rock.
You step into a stone object or surface large enough to fully contain your body
That means, a single stone object, large enough to fully contain your body. The DM could, of course, rule otherwise, and tell you that sure, you can meld into a "surface" comprised of sand, but to me the plain reading is a single solid stone object, at least large enough to hold you.
Does it have to be solid stone all the way to contain you or should it be just the surface large enough for you to enter? Example - stone road is not exactly deep enough to contain human but it's surface is large enough for human.
large enough to fully contain your body
That means, a single stone object, large enough to fully contain your body. See previous question.
What is visual of entering - is it like the stone is swallowing you slowly or, as the spell is instant, you are just entering stone as if you would move through opened door?
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.
While in stone you can cast spells with target self - does it include scrolls?
You . . . can cast spells on yourself
The source of the spell you cast is not defined. As long as "you" are "casting a spell", I think it works, to include scrolls.
However, there's one huge problem: can you see? The spell says:
you can't see what occurs outside [the stone]
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. 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.
Can you use your equipment abilities while in stone?
No. You can 1) cast spells on yourself 2) use your movement to leave the stone. "You otherwise can't move." Up for debate is whether you can even retrieve a scroll or material components. A lenient DM is likely not going to fret about details like that, but using any equipment seems off the table.
Ritual casting when moving - at which point you pick a target? At the end of the cast or it just fails if you have moved away? I would assume at the end.
This answer says "No explicit RAW, but evidence suggests target is selected when spell is released".
It seems reasonable that you cast the spell, then at the end you touch the stone.
That seems reasonable, but really it's something the DM will need to narrate on a case-by-case basis. For instance, if you're thinking of some tricksy thing where you ritual cast it and then pick the target at the very last moment to some great mechanical advantage (I can't think what, but maybe you've come up with something clever), then as a DM I'm going to be looking at it with a skeptical eye.
"You can use your movement to leave the stone where you entered it, which ends the spell" - do you leave it to the same point you casted the spell from or just exit the stone, doesn't matter where it was moved in the meantime?
To me, the spell is written as if assuming the stone does not move in the meantime. To me, it would be strange if someone could move the stone, and then you pop out and you're back where you started, but the rules don't say.
As a DM I would rule that in general, your exit is probably relative to the stone. I'm a fan that there are some things characters need to learn in game. Edge cases here seem like that sort of thing.
Can Distant Spell and Meld into Stone work together? - I believe it meets all criteria for this combo (distance: touch, so instead of touch just fill it out with "within 30 feet") but wanted to make sure if you agree.
It looks to me like it would. A pretty interesting combination, but in the end, what does it really buy you? In combat, sure, you could jump into a stone over there, but then what? Misty step is cheaper and more versatile, and doesn't require metamagic. Out of combat, well, you can instead just . . . walk over to the stone.
This is going to be highly DM-dependent. Nobody the Hobgoblin says in their answer you'd need to use movement to move the 30 feet. That seems reasonable. It also seems reasonable that it would be a 30-foot teleport.
If the DM rules you need to move, why even bother with the metamagic? Just move and cast.
If the DM rules it's a 30-foot teleport, well, at least you can escape opportunity attacks that way. I don't even think that's all that overpowered for the cost of the metamagic specialization and the sorcery point. But then you're stuck in a rock. A knowing enemy could play all sorts of havoc then.