Yes, you can move the image out of the original range
"Within range of the illusion" refers to the 20 ft cube illusionary object or creature, NOT the 120 ft casting radius.
As long as you are within 120 ft radius of the illusionary object, you may move it to any other point within range. This lets illusionists keep one persistent Major Image for any amount of time if they can keep using actions to keep it near them.
What defines "any other point within range" however, is up to discussion. It could mean you may move the object either 120 ft within the caster, or 120 ft within the illusionary object.
1. Can I create darkness with Major Image? Like a dark fog cloud that
me and my group can hide in?
Major Image (PHB p258)
You create the image of an object, a creature, or some other visible phenomenon that is no larger than a 20-foot cube.
This is a visible phenomenon so yes you can.
2. If my team knows that it's an illusion, can they see through the
darkness?
Major Image (PHB p258)
If a creature discerns the illusion for what it is, the creature can see through the image, and its other sensory qualities become faint to the creature.
If each character discerns the illusion for what it is then yes they can. It is up to the DM to decide what discern means. If they are forced to figure it out for themselves it requires an action to examine it and they must succeed on a Int(investigation) check against the DC of the spell. If they are told what it is by the caster then the DM could rule they automatically succeed on this check or perhaps have to make it with with advantage as it they had been aided. If it is a standard tactic by the party and they know it is an illusion (as asked in the question) it may not even need an action as it they already have discerned it for what it is. However as already stated it is up to the DM to determine what discern means.
So yes if your team knows, has discerned it to be an illusion (as determined by the DM) they can see through it.
3. Do foes need to decide to make an Intelligence check in order to
realize the illusion or they do the check anyway?
Physical interaction with the image reveals it to be an illusion, because things can pass through it. A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC.
This paragraph assumes you have made an illusion of a physical object. However your "dark fog" is not physical, things are supposed to pass through it. The intent of this description is clearly to say that if a creature is given a reason to know that it is an illusion they are seeing, then they see through it. The fog described is much harder to discern compared to a brick wall and so would probably require the creature to examine it as detailed in the spell.
The spell specifically requires a creature to take an action to make an Int(Investigation) roll, passive Investigation is not enough. There is no solid component to the illusion to reveal it's non-existance physically and the illusion provides all the other required components:
it seems completely real, including sounds, smells, and temperature appropriate to the thing depicted.
So yes the foes have to decide to take an action examine the dark fog or be given some other strong reason to know it is an illusion (the first one to realise screams "it's an illusion!" for instance).
4. If I create a dark fog around my enemies, are they blinded? (as long
as they don't know it's an illusion)
If the illusion is of something they cannot see through and they don't know it is an illusion then they cannot see through it. If the illusion was of a brick wall they would not be able to see what was on the other side. The same it true of this dark fog.
So yes they are blinded as long as the illusion is in effect and they are within it's area of effect. Foes outside the area of effect are not blinded of course, but cannot see into the area of effect.
It's a very potent way to use Major Image, but then it is 3rd level and a Fireball is pretty potent too.
Best Answer
Disclaimer: I have no experience with this in a gaming context, but I do work in a visualization research lab that gives me some insight into what this entails.
Binocular (i.e., two-eyed) perspective and depth perception only works in the first place because the world is highly consistent in its geometry. The human visual system, from the pupil, through the retina, and up multiple sections of the brain absolutely requires these consistencies to interpret what it is seeing. When I move ten feet forward, the pattern of light striking my eyes from different objects changes in highly consistent ways that let me infer-- with very high confidence-- what the positions of those objects are.
I mention this because one of the weaknesses of forced perspective illusions is that they in some sense depend on the "accident" of a person standing at just the right spot to create the illusion, and the farther you get from that spot, the less effective the illusion becomes until it just falls apart. (Note that I have seen this with my own eyes using virtual reality equipment. It doesn't take very many steps away for perspectives to get messed up, even when forced perspective illusions are not the subject being modeled. I have seen people in shared environments argue over the size of virtual objects purely because they were drawn precisely for the perspective of Subject A, while Subject B was standing five or ten feet away.)
And as soon as the brain detects these anomalies, it will almost involuntarily focus attention on them until the anomaly is understood. Usually this will involve moving the head, craning the neck, walking around something, etc. Human brains don't like inconsistent visual data.
So if we think about this in terms of virtual reality, that brings to mind the following very probable limitations:
If the image cannot move to respond to the movement of a viewer, it will almost certainly fail (or at least grant some significant mechanical bonus to the viewer on their Investigation check.) But this is not automatically insurmountable, because the image can move... as long as the caster is within 120 feet.
For this type of application specifically, affecting more than one creature at one time will be difficult. It is very hard to convey in words how rapidly perspective anomalies add up, but they do, even over a few paces. I would interpret this as mounting bonuses with additional numbers of viewers.
The more freedom someone has to walk around the illusion, or the more it depends on great distance, the faster it will break down. This makes things like "The floating castle above a hill" or "a wall a flame across a broad plain" are really hard to pull off-- when the subjects move, the illusion will inevitably break down.
As a corollary, in enclosed spaces, this technique might be highly effective. The illusion of a bottomless pit passes my smell test, for instance (subject to 2 above.) The illusion of a vast cave containing a great dragon at the end of a passage, likewise. Possibly even the castle on a hill or wall of flame if seen from inside a building through a single window.
Do not try this against ettins, Tiamat, or other multi-headed monsters. They will eat you.
The real answer to what's possible depends on a bunch of trigonometry for each application. But the five points above, I think, are a useful set of guidelines for what's possible, and what will keep your players or your GM from crying foul.
(Side note: The acoustic interactions, if any, will suffer the same basic problems but, for humans, much less severe. I would not assign a penalty to an illusion that incorporates an acoustic effect, because it would not be more difficult for the caster and probably would not help the subject pierce the illusion.)