Yes, that's almost (but not quite) correct.
Creating the wall of fire as a ring means everyone inside is going to take damage, yes. However, it's better than you're thinking, because they don't get any saving throws!
The saving throw is only for anyone in the area of the spell when the wall of fire first appears:
When the wall appears, each creature within its area must make a Dexterity saving throw.
That's the only time the spell grants anyone a saving throw. Since spells only do what they say and no more or less, and only grant saves when they say they do, nobody gets a saving throw for any other part of the spell.
Further, the area doesn't include the space surrounded by the wall: the area of a wall spell is only those spaces the wall itself actually occupies — everything surrounded by the wall is just “a bad place to be” (especially for wall of fire). And notice that the save is Dexterity: it's for partially getting out of the way of the forming wall, not for partially resisting the damage (like a Con save would indicate).
So the short version: you cast the spell as a ring with the damage side in, and everyone standing exactly where the wall forms takes 5d8 damage or half on a Dex save; meanwhile anyone surrounded by the ring takes no damage immediately, but will start taking 5d8 damage (no save) at the end of their turn, every turn, until the spell ends or they exit through the wall (taking another 5d8 fire damage, no save, for the effort).
One last point: Careful Spell metamagic (which someone mentioned in a comment) isn't very useful for a ringed wall of fire. All Careful Spell lets a sorcerer do is let a chosen creature automatically succeed on any saving throws the spell grants — it won't protect anyone surrounded by the ring, because they're not getting any saving throws. You could give a friend who happens to be standing right where you want the wall itself to form an automatic save (for half 5d8 damage) just when the wall comes into existence, but that's it.
Yes, but they don't stack.
PHB 206:
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap.
For your specific case, having two Walls of Fire in the same space does not double the damage. However, if it was a Wall of Fire and a Cloud of Daggers, those effects would overlap and targets in the area would take damage from both.
To answer the title question, nothing prevents the second Wall of Fire from being cast on top of the first, but it wouldn't add extra damage.
Best Answer
Strict RAW can be read this way but is confusing
A big part of the issue here is that the phrase "side of the wall" is not well-defined. It can refer to both an undefined area created by the wall's presence and also the physical surface of the wall itself. Both readings cause issues for this spell, RAW.
Side of wall = area reading
This is the way most people seem to want to read the spell (myself included). Unfortunately, a close reading reveals that reading it this way actually makes no sense with the way they chose to word it.
The problem with reading the phrase as being indicative of the area created by the wall's presence and not the physical surface of the wall is that it makes the phrase "within 10 feet of that side" make no sense. How can you be within 10 feet of an area that isn't even defined? Reading it this way, the spell doesn't even seem to work since there is no way to adjudicate where the effects are. Because it makes the spell not work at all, this cannot be the correct reading.
Side of wall = physical surface reading
Using "side of the wall" to mean the physical surface of the wall makes this clear. However, it then creates an issue that the area on the other half of the wall is technically within range of 10 feet of the surface of the wall.
Nothing in the description says that the other side dealing no damage strictly is meant to cancel out the clause in the first sentence which says that damage is done "within 10 feet of the [damaging] side". And technically 9 feet of non-wall space on the "non-damaging" side of the wall is within that range.
That leaves a narrow 1 foot segment on the "non-damaging" side that a creature would not take damage on.
The RAW is unintuitive and confusing, and I've never seen it played that way
While by a very strict RAW reading the above ruling could be called correct, I do not think it would be wise to run it that way. At the very least, I would not and have never run it that way and I think doing so would be a bad idea at most tables.
Firstly, by the RAW reading the phrase saying the other side deals no damage has almost no use whatsoever. It creates a small 1 foot gap of safety and that is it. This is not even enough for a creature to stand in. This seems unlikely to have been the intent of the spell (though I have no proof of this). If it had been intended, the designers could have written this so much easier by omitting the safe side verbiage entirely and having damage radiate from the wall 10 feet equally in both directions.
I do know that when players and DMs (every one that I have played with) read that they can make one side of the spell "[deal] no damage" they rightly expect that that side of the wall is safe for them to be on (eg "creatures on this side of the wall take no damage").
I think this is a good, natural reading for the spell and I think the RAW is straining quite a bit. Running it by RAW is a jarring diversion from this reading and this could cause confusion and arguments at the table.
The RAW also would increase the damage potential of the spell by quite a bit. Now the damaging area of the spell is almost double. This does come with the tradeoff of making it much harder for the players to use the wall for protection on one side of them in cramped areas (without taking damage themselves). It also makes it more difficult to avoid damaging allies.
Personally, I think that the RAW should be disregarded here in favor of the more common and natural reading of the spell, but your table should do whatever is the most fun for them.