How did the Time gem work in Dormammu's dimension?

The answer here is the key revelation that Strange comes to in the climactic scene: he takes time with him into Dormammu's domain.

To understand this, we have to backtrack to the Ancient One explaining to Strange how magic works. She explains that there are many other dimensions, each of which has their own set of rules. Many of these dimensions do not follow our natural laws -- Dormammu's domain, for example, has no concept of "time". What exactly this means isn't clear, but the implication is that Dormammu, and those things in his realm, do not "age" or "evolve" or even "change", as those are all concepts associated with time. His domain exists and grows by absorbing other dimensions, at which point those other dimensions become eternally part of Domammu's realm.

To cast spells, sorcerers draw energy from those other dimensions and pull it into our world, allowing them to "break the rules" using that extra-dimensional energy. What Strange figured out was that he could do the same thing, in reverse.

By using the Eye (in reality, the Time Gem), he could travel to Dormammu's dimension and draw "time" from our reality, and use it to do magic. The Eye allowed him to, in essence, pull a "bubble" of time with in, that affected himself and the area around him. Since time is not a natural part of that world, Strange (via the Eye) had total control over how it worked, just like sorcerers in our world have total control over magic.

This is why Dormammu was basically helpless against Strange's attack: Dormammu existed outside of time, and had no way to affect it or understand it; much like the average mortal in our world has no concept of magic or how to use it. So Dormammu was "stuck" inside Strange's spell, and only agreeing with Strange to break the spell let him escape.

[Note: Speaking only from personal experience with the movies and nearly no experience with the comics whatsoever.]

Throughout the Marvel movies, these "Infinity Stones" seem to be named for, and to reflect upon, their nature of being of infinite power and effect.

Mathematically speaking, infinity can be conceptualized as the inverse of nullity (nothingness). Formulaically, we describe null as 0 and the inverse of zero would be infinity: 1/0. Since x/0 has no algebraic solution, the De L'Hôpital approach would treat the denominator as the width of a differential. Dividing a differential of any size (dx) by a null differential (d0) is essentially equivalent to driving the denominator towards nullity. You cannot really divide by nothing, but we can conceptually speak of what it means to divide by "something" that is approaching a dimensional-metric (a width) of "nothingness". All in all, this is the concept of a Singularity.

In our reality, we perceive of a 4-dimensional space-time. Our three spatial dimensions (height, width, and depth) vary along the unidirectional dimension of time; meaning that we only ever seem to go forwards in time. It's like time is a force, like gravity, and our universe is falling under its force, and we can't fight against it to go backwards. But if someone is pushing on you with a force, though you may not be strong enough to push back, that doesn't mean you can't go around them.

If reality, beyond our perception, is hyperdimensional to our 4-D perceptions, then there exist fifth and sixth dimensions and beyond, such that if we could move along those dimensions we could go around time and essentially end up forwards or backwards in time.

In this concept of dimensionality, a singularity -- an infinitude -- is a point of convergence, where all (or some) dimensionalities reach their limit. Dimensionality, for perceptual understanding and practical considerations, requires an origin, a limit, and a progression -- meaning that dimensionality must consist of a "metric space". As a measured dimensionality -- a metric space -- there is a nullity and an infinity. All points are measures from the nullity towards the infinity. If we picture our 3-D space as a sphere, the surface of that sphere is the infinity of our space, with the nullity (origin) at the center of the sphere. Infinity in this sense can be positive or negative, and has directionality in relation to the 3 components (call them x, y, and z) of our space. You can be at infinity in one, all three, or some combination of the dimensions, but you always end up on the sphere.

In other conceptual constructions of a multi-dimensional space, like a Projective Space (the spherical model is called Euclidean or Cartesian Space), there is only 1 infinity, a singular point of infinity, and the origin is actually a point, disc, sphere, or hypersphere. This is mostly conceptual, but there's a reality to it in the sense of modelling optics and light and abstract dimensional-scaling. In the Projective Space, the point at infinity is a singularity, as all infinities are, and it behaves like a "black hole". You can conceptually map points through the point at infinity, linking positive points to negative points, and the "sign" (negativity or positivity) of a point takes-on a meaning of directionality. By mapping through infinity, you create inversions, you "flip" every direction. So if a body, a collection of points, could pass through the singularity at the point of infinity, it would come out the "other side" inverted -- inside-out and upside-down, in every dimensionality of the space.

If we go back to Euclidean space, what happens when hit the surface of the sphere, but then keep travelling? Do you end up moving back towards the origin from where you hit -- did you bounce off the sphere? Or do you move towards the origin from the exact opposite side of the sphere? Which would be similar to Futurama episode (S06E07, "The Late Philip J. Fry"), where they go so far forward in time that they "come out the other side" and "catch up" with where they left off.

This is the concept of the "black hole" or the "worm hole". Singularities, points in dimensional space-time, that are connected to other points and allow for "teleportation". More conceptually, the singularity, leading back to the divide-by-zero issue, is a logical fallacy. How can you have a metric space, but be free to teleport from within the metric space? How does that not break all the concepts and reality of the measure(s) between discrete points in the space?!?

It does. But what if the space, the 4-dimensions, existed inside of a higher dimensional reality?

Now imagine that there is a 13-dimensional reality, and space-time is only 4 of those 13 dimensions. What appears like a singularity, a divide by zero, in our 4-dimensions (ex: [1, 1, 1, 1] / [0, 0, 0, 0]) is only such because it is null in our 4 dimensions but is non-null in other dimensions (ex: [1, 1, 1, 1, 1] / [0, 0, 0, 0, w]). We still have a divide-by-zero issue, but going back to (De L'Hôpital), that means that it's just beyond our language; we aren't really dividing by zero, we're heading towards an inverse nullity, which is really an infinity. But that means we're only heading towards infinity in our dimensions. In other dimensions we could be anywhere. So we might be at infinite time and infinite space points, but what if gravity has a dimensionality, a scale of its strength, we're at the +9.8 point of gravity, so what if we "circumvent" time by going to the -10.2 point of gravity then come back around to a "previous" point in time?

So, an "Infinity Stone", conceptually, seems to exist as a singularity point, a point at infinity, for its dimensionality. The "Time Gem", exists as the singularity for time, but allows free travel -- as the holder is so able to do -- in all other dimensions. Because you can travel freely -- in the limits of your abilities -- in all other dimensions, you can circumvent time. It's like time is a line drawn down the center of a room. If you travel along the line, you can't ever escape it, you're moving only forwards or backwards in time. But you're in a room, a room of many dimensions beyond this line. Imagine jumping over the line, running around the line, falling onto the line... you can go around it however you want, so long as you leave the line. But if you've left the line, where are you "on" the line? How did you "get off of the line"? Leaving the line, leaving "time", you've jumped out of the dimensionality, you've escaped into other dimensionalities, you've gone past the origin and the inverse origin, you've gone through the singularity. You are at "infinity" in time, meaning that you are anywhere except at a point in time; you're not at the beginning or the end, you're beyond. All things that are beyond time are immeasurable and innumerable by time, so they all "exist" at a singularity, at the infinity point "in" time.

And so, Dormammu is in a "Dark Dimension" -- this is bad grammar and should be referred to as a "Dark Space" of at least 3-dimensions, because we see at least 3-dimensions during the movie; it's clearly not a single "dimension". The space of Dormammu, the dimensionality seems to have at least our similar 3 spatial dimensions, but as a space "beyond time", that is a grammatically tolerable way of saying that there is no "time dimensionality" to that space. If there's no "time", then things can move, but they aren't moving in time, they're moving either purely spatially, thus never changing/evolving temporally but rather becoming new spatial points, being destroyed and recreated as something technically different with every "motion" -- which seems pretty "dark" and f***ked up, right? Or there are other dimensions, like gravity or electromagnetism or things we have no concepts for, or maybe even "Love" is a dimension; right, "Interstellar"?!?

Either way, it's a space without a time dimension, that's what it means to be "beyond time". The "time gem" is a singularity of time, an infinity, but that means that it exists in all other possible dimensions, so even if Dormammu's space has no time dimension, Dr. Strange can use the gem to travel "around time" in some other dimensions -- whichever ones he's comfortable traversing through -- and impose a "time loop" on the reality space that Dormammu exists within. Dormammu is stuck because the Time Gem is a worm-hole allowing Dr. Strange to travel through other dimensions, even "beyond" the point-in-time of his death. So when Dormammu kills him, he just returns to the "checkpoint" in time that he registered himself to with the "spell" cast using the Time Gem and the "magical artifact" it's held inside of. Now, this would mean that Dormammu could have saved itself by stealing the Time Gem from Dr. Strange and using it itself to go "around" the time loop; but clearly it didn't understand what was happening.

And this is the looming threat of collecting all the Infinity Gems into a Gauntlet of Power. If someone ends up with all the Infinity Gems -- Soul, Time, Space, Mind, Reality, and Power -- they have control of 6 singularities among the N total dimensions of all of reality. For each Gem, that means they can freely "escape" that dimension and traverse the others to manipulate "reality" in that Gem's dimensionality.

This is probably why it's not a great idea to have a jerk like Dr. Strange walk around with it just hanging on his neck in a super obvious ornate piece of magical jewellery that all but screams "steal me!".