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.

They are "real images"

Illusions can create either a "real image" or sensory experience that anyone can perceive, or plant images directly in the mind of creatures.

The Schools of Magic, PHB 203

Illusion spells deceive the senses or minds of others. They cause people to see things that are not there, to miss things that are there, to hear phantom noises, or to remember things that never happened. Some illusions create phantom images that any creature can see, but the most insidious illusions plant an image directly in the mind of a creature.

This implies that "phantom images" created such that any creatures can see them (Silent Image, Programmed Illusion), are distinct from illusions that plant images directly in the mind of a creature (Fear, Phantasmal Force). This further implies that the images created by silent image are not planted directly in the mind of its observers.

Mechanical Perspective

The spells minor illusion, silent image, programmed illusion, and disguise self all have a range.

Let's take for example, the spell disguise self. It has a range of Self and can change the appearance of the caster superficially. These magical alterations, though limited only to the caster (as seen in the range of the spell), affects the perspective of anyone who can see it -- even from hundreds of feet away, if visibility allows.

Similarly, the spell silent image can create the image of an object within 60 feet of the caster, affecting an area no larger than a 15-foot cube by creating a visual phenomenon in that space. Therefore, the spell does not affect the minds of its observers (at least, not directly), but rather only a specific spot within range of the caster.

So, if someone 300 feet away looks at this image, they should still be able to see it, since they aren't being directly influenced by the spell. They're just perceiving a space differently, and that space is what this spell is affecting.

The same arguments can be used for programmed illusion and minor illusion. These spells create illusions that exist outside the mind of their observers.

The "paradox" of different people seeing different things

Once a visual illusion is discerned, it becomes faint to that creature and to that creature only. Meanwhile, others continue to see it as if it was really there. This imparts to these illusions properties of a mass hallucination, because two people can look at the same place and see different images (one sees a "full" image, the other sees a transparent one).

Strictly speaking, from a mechanical PoV, these images are external to the observers. So there should be an in-world explanation for why they have properties that suggest they are hallucinations instead.

This can be explained any number of ways, though, without being cornered into accepting the illusions as directly affecting the mind of the observers. First of all, it's magic, which means it doesn't have to be intuitive. One way to explain it is, since these illusions affect a space (not you directly), and you are just perceiving that spot differently, the Rules of Magic could change depending on your knowledge as the observer. There are other spells which have this property too: zone of truth does not allow you to deliberately lie, but you can say something that is false if you believe it to be true; geas could allow you to act against the direct instructions of the caster if you don't know you are violating those instructions (otherwise you could use geas to say "Say only things that are true about X" and everything the target says will have to be factually true about X, even if they had no way of knowing if it was true otherwise).

Another way, which is simpler, is to say the spell could be presenting you with an optical illusion, one whose nature is such that once you see it, it's hard to "unsee."

Implications

Antimagic Field

The spell says:

Spells. Any active spell or other magical effect on a creature or an object in the sphere is suppressed while the creature or object is in it.

This clearly means illusions that create images in the minds of its targets are absolutely nullified if the target enters an area affected by antimagic field. But they can still see an image (created by, say, silent image) that is not in that area, for ex., one just standing outside the border, because that image is not inside the area affected by antimagic field.

Detect Magic

The spell says:

For the duration, you sense the presence of magic within 30 feet of you.

If a magical illusion is within 30 ft of the caster, then the caster will detect it (but they won't necessarily know where it is, or which school it's from). If a phantom image that anyone can see enters the range of 30 ft, then they will sense magic is present. If someone affected by the fear spell enters their range, they will also detect the presence of magic.

Light

This is a messy topic, and is best left to DM adjudication.

If an illusion cannot block light, then it must be invisible -- you would see through it because, to see in the first place, you need to see the light coming off it. If it can't block light, then that means you are still seeing the light coming from the object directly behind it, and the illusion is covering nothing.

On the other hand, if an illusion can block light, then illusion spells which say: "[p]hysical interaction with the image reveals it to be an illusion, because things can pass through it" are contradicted, because light is a thing. If light, as a thing, can't pass through the illusion, then the clause on illusions being unable to block things is violated.

So the implications for light, as far as illusions go regarding "are they visible to all, or only in your head?" are messier and tangentially related at best. It is probably applicable only to spells which conjure phantom images that anyone can see, though, and not to illusions that plant images directly in your mind.

Vision

Line of sight is defined in the DMG:

Line of Sight, DMG 251

To precisely determine whether there is line of sight between two spaces, pick a corner of one space and trace an imaginary line from that corner to any part of another space. If at least one such line doesn't pass through or touch an object or effect that blocks vision -- such as a stone wall, a thick curtain, or a dense cloud of fog -- then there is line of sight.

If something blocks your vision, you do not have line of sight. If an illusion obstructs your vision -- ie, you cannot see what is behind that illusion -- then your line of sight is cut.

Notably, even a thick curtain can block line of sight. Thick curtains probably cannot shield you from an arrow or a beam created by disintegrate, but it can obstruct the vision of those who would use these things against you, and that is enough to stop them from targeting you.