Can I create darkness with major image? Like a dark fog cloud that me and my group can hide in?
If my team knows that it's an illusion, can they see through the darkness?
Do foes need to decide to make an Intelligence check in order to realize the illusion or they do the check anyway?
If I create a dark fog around my enemies, are they blinded? (as long as they don't know it's an illusion)
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.
Physical interaction with the image reveals it to be an illusion, because things can pass through it
This would be for either placing the image on someone or on yourselves as it does not specify that you, the caster, are immune to this side effect. So for these, No it would not work in this instance.
They would still need to make an Intelligence(Investigation) check
An enemy would still need to use his action to discern whether it's an illusion, the image is believable in all aspects until that action is taken to discern whether it's an illusion, or the previously mentioned side effects take place(physical interaction)
...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. 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.
The thing to consider is how an illusion acts compared to how the real thing acts. Physical objects are easily discerned because once you touch it, it’s not there. Mist also reacts when someone walks through it due to displaced air, so someone may notice the mist is not reacting to wind or passes through objects. Darkness itself may be harder to discern because of no physical component, but depending on what kind of darkness it is there are hints. If your imitating natural darkness, recall light will not react with it normally, if imitating the darkness spell, even that doesn’t block light like a darkness spell. Imagine a black sphere that is a darkness spell, you could say the real spell casts a shadow since it is actually blocking light, but the illusion does not. It could be as simple as having an enemy make a perception check to spot the detail followed by an investigation check to identify it as an illusion. The rule I tend to think of is that illusions power comes from people believing they are real, and it’s the dispelling of that belief that reveals an illusion for what it is.