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Level Two: Citadel of Osmet Khan, The Talisman
The first test was a trick (which you might've been able to guess, since the second one was as well). Chateau d'Alfar may well be "the finest wine in the world," a fact we learned from the voice at the chasm... but there were a couple of other facts to consider. The first occurred wayyyyyyyyyyyy back at the start of the level, when we dined with the merchant:
The dragon likewise told us that the Tercelidae have no taste for wine. So we can eliminate everything that's a wine, or seems like one. In fact, we can probably eliminate most plant-based alcohols, which narrows things down considerably. The right answer, of course, is kumiss.
Oh, what, you're not intimately familiar with the steppe nomad's fondness for fermented mare's milk? Granted, I may have lied just a little bit; the dragon actually tells us this directly on top of his other clue about the Tercelidae having no taste for wine, but I wanted to see if anyone would catch themselves and remember that we're selecting based on the tastes of a Tercelid. Even without that hint you can narrow it down to kumiss or water; everything else is grain or vine alcohol.
At least nobody got tricked by the second test! The answer, regardless of what we believe, is one.
Now for the Dreaded Al-Gibra. Back at the Twisted Temple we found a diagram purporting to reveal its secrets:
For the solution, let's go with nweismuller and Tenebrais's explantions:
nweismuller posted:
The diagram in the Burning River update helpfully gives us our method. Our target length is R. 24 fills in for the 2A in the diagram; the 8 we can measure is the distance below the horizontal line through the circle in the diagram. Let us define the remaining distance beyond what we can reach to the center of the circle as X. Thus, R = X+8. By the Pythagorean theorem, X^2 + A^2 = R^2; since 2A is 24, that translates to X^2 + 144 = R^2. Substituting R-8 for X, we get R^2 -16 R +64 +144 = R^2, which is equivalent to 16R = 208, or R=13. We want a 13-foot ladder.
Tenebrais posted:
We want to find R, and we know that 8 is contained within it so the remainder of the distance is x feet. R = 8 + x
The chord is 24 feet long, and half that distance is 12. 12, x, and R form a right-angled triangle as per the diagram in the river of fire.
So R2 = 122 + x2, and R2 = (8 + x)2 = x2 + 16x + 64
Combine these and x2 + 144 = x2 + 16x + 64
Cancel out some of this to get 80 = 16x
Divide out for 5 = x
Add the 5 to the original 8 and the ladder is thirteen feet long.
The math gets a little hard to follow toward the end there with the cancelling out, but it's correct. So the answer should be to use a 13 foot ladder. Let's see if the game agrees.
At last, the tunnel leads to a small chamber. There is a door on the opposite wall. The chamber has no windows; it is lit by globes of glass that glow dimly red. You look more closely, and see that within each globe, a spark dances. You hear a faint scream; you have heard of this before. The globes are lit by souls in torment, their agony powering the light. Before you leave, you vow, you will smash these globes.
As if we needed more evidence that Aboud is a dick. Perhaps these are the souls of other would-be Talisman thieves? Apparently Persephone was here, but failed to acquire the Talisman. She survived, so she must've been warped out by Aboud and chose a different route past the palace, for we know she made it at least to the third level. We, on the other hand, have succeeded where all others failed. Our prize is before us:
At the center of the room is a glass case; and within it...
Reverently, you open the case and lift the talisman. The wisdom of a hundred generations of shamen; the magic of a people. It was this small object that allowed Chugotai Khan to conquer all the lands between Carlon's kingdom and the MadMaze itself -- before they were swallowed up by the Mad One's domain.
The writers mistakenly pluralized shaman as "shamen;" the correct pluralization is "shamans." 
You will put it to better use. You use it to smash the globes.
Score one for us. We now have a magical artifact, which we immediately use as a blunt instrument! 
So... now what? Hrm. I guess we could go back the way we came, and look for a way to the third level.
You crawl back down the tunnel, clutching your prize. You come to the lip and peer down... The ladder is gone. It is thirteen feet to the ground -- a jump, but one you might survive. Do you care to take the risk?
Before we do, can the Talisman help somehow?
You clutch the Talisman tightly and close your eyes... Nothing happens. Feeling slightly silly, you open your eyes and peer down again. The only thing that has changed is that you have the distinct impression that jumping would be suicide.
Going to have to agree with our vaguely-magical insight here, that would be stupid. Let's try the door in the treasure room instead.
Oh of course, it opens onto a sheer drop into nothingness. Why wouldn't it? This seems like suicide too. But wait a second... there's a tapestry over there out of reach. You don't suppose...
You hold the Talisman tightly and close your eyes. You hear faint bass voices chanting in your mind... When you open your eyes again, the "tapestry" is floating in the air before you. You can step onto it easily.
The tapestry was a flying carpet, and the Talisman activated it for us. I think I'm going to enjoy this whole "having magical powers" thing.
You step onto the carpet... It bears your weight, but it simply floats in space. Now what?
Maybe the Talisman can make it move?
You close your eyes and concentrate. You hear someone making a sound of disgust. Nothing else happens.
Guess it won't solve all our problems for us. Well, if it's a flying carpet, there's probably some magic words that start it up. We only know one set of words for magic carpets...
...But sure enough, it's available as an option. Na Namblor Voyant!
The carpet accelerates from the cliff smoothly and quickly. You sit down for greater comfort. It's sailing over the desert now, ochres and yellow sands pass far beneath you...
Suddenly, you are flying through darkness. You feel the carpet beneath you, Valterre by your side; but you can see nothing, for it is pitch blank.
I assume they meant pitch black? I suppose it's blank as well.
...And then, silver light is all around you. And before you, a woman holds a book. No mere woman, but the Lady. And no mere book, but the Scroll of Heroes.
Congratulations! For clearing the second level, we've been given the right to record our names for all posterity... provided all posterity includes the time MadMaze was available on Prodigy, and also the time Prodigy was available. So this feature does nothing but make us feel accomplished. But we have done quite a lot, and the game's about half over, so let's pat ourselves on the back; we deserve it.
And as the pen forms the last letter of your name, the Lady, the silver light, the book, and even the pen disappear. It is broad daylight, now, and your carpet flies over the desert still...
You espy it ahead and below you. But the Third Level is bizarre. The walls follow no obvious pattern: they form strange angles, intersect in weird ways. The shapes are hard for the human mind to grasp... And you can see what must be Places of Power, but they are even stranger. Dark, misshapen things are there; and geometries beyond human ken.
The third level still has square mazes. Sorry to burst the narration's bubble.
With trepidation, you fly onward... You gulp with apprehension, but you have not come this far to be deterred from your goal.
This final screen is a bit of a joke. MadMaze was released level by level, and the paths to the next were blocked off until the game was updated. Had we arrived here many decades ago, we might be forced to wait while a troll work crew finished up access, and our journey would temporarily end here. Fortunately we'd receive a verse from the Cipher so that we could restart immediately, and... hey wait. We didn't get any Cipher verse.
Regrettably, they sort of abandoned the whole Cipher thing (the last one we could use would put us at the start of the second level), but there actually is a verse to access the third level! Apparently it was given to you if you finished the second level before the third was open, but it's nowhere to be found in MadMaze-II's code or resources. By sheer luck, I found it on a Prodigy forum archive:
The bonds of time
Lie in the past.
The Argent Lass
A garden hoes.
If we, blind, force us to see
Then yarely bless
our loins gird:
Fight fervently for truth.
And that's it! In a little under a month, we've reached the third level of the MadMaze. There's a lot to go over going forward, including intriguing new features such as access to a (limited) magic system through the Talisman of Chugotai. Our options for solving puzzles have grown even beyond "Valterre means you can kill stuff sometimes." But of course there are things that cannot be easily killed by magic blade or magic magic, so don't get cocky.
For you see, the third level is weird, as appropriate for a place long within the Mad One's domain. No longer do we tread in recognizable mythology. No King Arthur, no Arabian Nights. Fortunately this means any parallels to real world stuff will also be largely absent, so hopefully there'll be nothing terribly offensive ahead (EDIT: I checked; there is). Except the puzzles, perhaps; at least one of them is genuinely infuriating.
Alternate Solutions & Deaths
Turns out the chasm's hint about Chateau d'Alfar was a trick, and the game reacts specifically to its selection.
Choosing it, or anything other than kumiss, ends up failing the test:
There is a howl as of a demon in pain. Sudden light flashes through the basement chamber. The wine rack seems to grow; it is as high as a palace, as a mountain. It topples. You turn to flee, but are not fast enough; it falls and crushes you like an insect. You feel the life fleeting from your broken form.
For the second test, there's of course a special answer for claiming that there are nine gods.
You know that your answer is true; there are nine spirits known as gods in this world, the Mad One among them. This is not, however, the answer the demon wants.
That's a real pisser. On to the Dreaded Al-Gibra. Should we choose too short or too long a ladder:
The ladder is too (short/long). It touches the blue flame -- Lightning crackles down the length of the ladder. It turns you into a grease spot.
That's not the only way the Al-Gibra can kill us, however. Should we decide to crawl back down the tunnel after acquiring the Talisman, we can attempt to jump out from its maw. "Attempt" being the key word here, as it will not let us escape so easily.
Finally, we can die in numerous ways via the door to nowhere. Of course, walking out is one way:
You step out onto the air... and begin to plummet.
But we can also activate the carpet, get on it, and dawdle doing nothing or speaking the wrong words. After too long...
Suddenly, the carpet begins to fall...
In either case, we die horribly in what appears to be outer space or something:
That's strange. Did we fall off the edge of the world? Does the world even have an edge? And if it does, why would it be here of all places?
Just gonna go with "Mad One" on this one. It perfectly non-explains everything.