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.
Ask your DM
Unfortunately, there is very little guidance in the books on how much you can do or can't do with illusions outside of the general guidelines in each spell.
We do know that within Initiative Order, your action and turn take up about 6 seconds. What your action can do with regard to directing the image needs to fit into this timeframe.
With regard to spell description section you quoted above, this is not a live action and response effect. You are using your action to program how the illusion will look and sound. It may not be able to respond in real time depending on your table treats conversations during Initiative Order, but you could have it appear to be in a conversation (most likely with something else in the illusion, but if you're very clever you may be able to make it appear to be responding with generalities. The key difficulty is in that you are using your Action to make it respond. If the other party in communication goes after your turn, you may not be able to respond - but again, that's a table dependent issue.
Best Answer
There is no restriction that the illusion cannot occupy the same space as a creature (it is not a creature itself and the text does not mention it). You can also create the image of a creature you can see easily and slight modifications, like the lack of wounds should be within the capabilities of the spell. The only hiccup could be with moving the illusion to overlap with the creature at all times. The spell does not state that the image can move by itself, only says that:
The precise details will be up to the DM, but it seems reasonable to rule that if you spend your action every turn on moving the illusion, it can stay overlapped with the creature.
Limitations
Unless you are a high enough level wizard of the school of illusion, you cannot change the image after you have cast it. You cannot make only some wounds appear.
You can only cover things up, you cannot make anything disappear. Like possible sprays of blood from a really damaging slashing attack.
About the image giving orders
Unless the real creature is mute and/or deaf, he will also hear the things you make "him" say. It might give away your trickery sooner. Also, he can just override your "orders" with his own. You might create some confusion in the ranks, but then some of them will surely make investigation checks to clear it up.